Leveraging graphical model techniques to study evolution on phylogenetic networks

Abstract

The evolution of molecular and phenotypic traits is commonly modelled using Markov processes along a rooted phylogeny. This phylogeny can be a tree, or a network if it includes reticulations, representing events such as hybridization or admixture. Computing the likelihood of data observed at the leaves is costly as the size and complexity of the phylogeny grows. Efficient algorithms exist for trees, but cannot be applied to networks. We show that a vast array of models for trait evolution along phylogenetic networks can be reformulated as graphical models, for which efficient belief propagation algorithms exist. We provide a brief review of belief propagation on general graphical models, then focus on linear Gaussian models for continuous traits. We show how belief propagation techniques can be applied for exact or approximate (but more scalable) likelihood and gradient calculations, and prove novel results for efficient parameter inference of some models. We highlight the possible fruitful interactions between graphical models and phylogenetic methods. For example, approximate likelihood approaches have the potential to greatly reduce computational costs for phylogenies with reticulations.

Full text

LEVERAGING GRAPHICAL MODEL TECHNIQUES TO STUDY
EVOLUTION ON PHYLOGENETIC NETWORKS
Benjamin Teo
Department of Statistics
University of Wisconsin-Madison
Paul Bastide
IMAG, Universit´
e de Montpellier,
CNRS
C´
ecile An´
e
Departments of Statistics and of Botany
University of Wisconsin-Madison
ABS TRAC T
The evolution of molecular and phenotypic traits is commonly modelled using Markov processes
along a rooted phylogeny. This phylogeny can be a tree, or a network if it includes reticulations,
representing events such as hybridization or admixture. Computing the likelihood of data observed at
the leaves is costly as the size and complexity of the phylogeny grows. Efficient algorithms exist for
trees, but cannot be applied to networks. We show that a vast array of models for trait evolution along
phylogenetic networks can be reformulated as graphical models, for which efficient belief propagation
algorithms exist. We provide a brief review of belief propagation on general graphical models, then
focus on linear Gaussian models for continuous traits. We show how belief propagation techniques
can be applied for exact or approximate (but more scalable) likelihood and gradient calculations,
and prove novel results for efficient parameter inference of some models. We highlight the possible
fruitful interactions between graphical models and phylogenetic methods. For example, approximate
likelihood approaches have the potential to greatly reduce computational costs for phylogenies with
reticulations.
Keywords belief propagation, cluster graph, admixture graph, trait evolution, Brownian motion, linear Gaussian
Contents
1 Introduction 3
2 Complexity of the phylogenetic likelihood calculation 3
2.1 Thepruningalgorithm .......................................... 3
2.2 Continuous traits on trees: the lazy way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 BPforcontinuoustraitsontrees ..................................... 5
2.4 Fromtreestonetworks .......................................... 5
2.5 Current network approaches for discrete traits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.6 Current network approaches for continuous traits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Continuous trait evolution on a phylogenetic network 6
3.1 LinearGaussianmodels.......................................... 6
3.2 Evolutionary models along one lineage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Evolutionary models at reticulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.4 Evolutionary models with interacting populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
arXiv:2405.09327v1 [q-bio.PE] 15 May 2024

Leveraging graphical model techniques to study evolution on phylogenetic networks
4 A short review of graphical models and belief propagation 8
4.1 Graphicalmodels............................................. 8
4.2 Phylogenetic examples of graphical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 BeliefPropagation ............................................ 11
4.4 BPforGaussianmodels.......................................... 13
5 Scalable approximate inference with loopy BP 15
5.1 Calibration ................................................ 15
5.2 Likelihoodapproximation ........................................ 16
5.3 Scalability versus accuracy: choice of cluster graph complexity . . . . . . . . . . . . . . . . . . . . . 16
6 Leveraging BP for efficient parameter inference 19
6.1 BP for fast likelihood computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.2 BP for fast gradient computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.3 BP for direct Bayesian parameter inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
7 Challenges and Extensions 21
7.1 Degeneracy ................................................ 21
7.2 Loopy BP is promising for discrete traits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
A Recasting SnappNet as BP 31
B Bounding the moralized network’s treewidth 32
C Approximation quality with loopy BP 33
D Gradient and parameter estimates under the BM 34
D.1 ThehomogeneousBMmodel....................................... 34
D.2 BeliefPropagation ............................................ 34
D.3 Gradient computation and analytical formula for parameter estimates . . . . . . . . . . . . . . . . . . 38
D.4 Analytical formula for phylogenetic regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
E Regularizing initial beliefs 41
F Handling deterministic factors 43
F.1 Substitution................................................ 43
F.2 Generalizedcanonicalform........................................ 44
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Leveraging graphical model techniques to study evolution on phylogenetic networks
1 Introduction
In phylogenetics, data are observed at the leaves of a phylogeny: a directed acyclic graph representing the historical
relationships between species, populations or individuals of interest, with branch lengths representing evolutionary time
and internal nodes representing divergence (e.g. speciation) or merging (e.g. introgression) events. Stochastic processes
are used to model the evolution of traits over time along this phylogeny. In this work, we consider traits that may be
multivariate, discrete and/or continuous, with a focus on continuous traits. Inference from these models are used to
infer evolutionary dynamics and historical correlation between traits, predict unobserved traits at ancestral nodes or
extant leaves, or estimate phylogenies from rich data sets.
Calculating the likelihood is no easy task because the traits at ancestral nodes are unobserved and need to be integrated
out. This problem is very well studied for phylogenetic trees, with efficient solutions for both discrete and continuous
traits. Admixture graphs and phylogenetic networks with reticulations are now gaining traction due to growing empirical
evidence for gene flow, hybridization and admixture. Yet many methods and tools for these networks could be improved
towards more efficient likelihood calculations.
The vast majority of evolutionary models used in phylogenetics make a Markov assumption, in that the trait distribution
at all nodes (observed at the tips and unobserved at internal nodes) can be expressed by a set of local models. At the root,
this model describes the prior distribution of the ancestral trait. For each node in the phylogeny, a local transition model
describes the trait distribution at this node conditional on the trait(s) at its parent node(s). As each local model can be
specified individually with its own set of parameters, the overall evolutionary model can be very flexible, including
possible shifts in rates, constraints, and mode of evolution across different clades. Other models do not make a Markov
assumption, such as models that combine a backwards-in-time coalescent process for gene trees and forward-in-time
mutation process along gene trees. We show here that some of these models can still be expressed as a product of local
conditional distributions, over a graph that is more complex than the initial phylogeny.
These evolutionary models are special cases of graphical models, also known as Bayesian networks, which have been
heavily studied. The task of calculating the likelihood of the observed data has received a lot of attention, including
algorithms for efficient approximations when the network is too complex to calculate the likelihood exactly. Another
well-studied task is that of predicting the state of unobserved variables (ancestral states in phylogenetics) conditional on
the observed data. We argue here that the field of phylogenetics could greatly benefit from applying and expanding
knowledge from graphical models for the study and use of phylogenetic networks.
In section 2 we review the challenge brought by phylogenetic models in which only tip data are observed, and the
techniques currently used for efficient likelihood calculations for phylogenetic models on trees and networks. In
section 3 we focus on the general Gaussian models for the evolution of a continuous trait, possibly multivariate to
capture evolutionary correlations between traits. On reticulate phylogenies, these models need to describe the trait of
admixed populations conditional on their parental populations. Turning to graphical models in section 4, we describe
their general formulation and show that many phylogenetic models can be expressed as special cases, from known
examples (on gene trees) to less obvious examples (using the coalescent process on species trees, or species networks).
We then provide a short review of belief propagation, a core technique to perform inference on graphical models, first in
its general form and then specialized for continuous traits in linear Gaussian models. In section 5 we describe loopy
belief propagation, a technique to perform approximation inference in graphical models, when exact inference does not
scale. As far as we know, loopy belief propagation has never been used in phylogenetics. Section 6 describes leveraging
BP for parameter inference: fast calculations of the likelihood and its gradient can be used in any likelihood-based
framework, frequentist or Bayesian. Finally, section 7 discusses future challenges for the application and extension of
graphical model techniques in phylogenetics. These techniques offer a range of avenues to expand the phylogeneticists’
toolbox for fitting evolutionary models on phylogenetic networks, from approximate inference methods that are more
scalable, to algorithms for fast gradient computation for better parameter inference.
2 Complexity of the phylogenetic likelihood calculation
2.1 The pruning algorithm
Felsenstein’s pruning algorithm Felsenstein [1973, 1981] launched the era of model-based phylogenetic inference, now
rich with complex models to account for a large array of biological processes: including DNA and protein substitution
models, variation of their substitution rates across genomic loci, lineages and time, and evolutionary models for
continuous traits and geographic distributions. The pruning algorithm gave the key to calculate the likelihood of these
models along a phylogenetic tree, in a practically feasible way. The basis of this algorithm, which extends to tasks
beyond likelihood calculation, was discovered in other areas and given other names, such as the sum-product algorithm,
message passing, and belief propagation (BP).
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Leveraging graphical model techniques to study evolution on phylogenetic networks
The pruning algorithm, which is a form of BP, computes the the full likelihood of all the observed taxa by traversing the
phylogenetic tree once, taking advantage of the Markov property: where the evolution of the trait of interest along a
daughter lineage is independent of its past evolution, given knowledge of the parent’s state. The idea is to traverse the
tree and calculate the likelihood of the descendant leaves of an ancestral species conditional on its state, from similar
likelihoods calculated for each of its children. If the trait is discrete with 4 states for example (as for DNA), then
this entails keeping track of 4 likelihood values at each ancestral species. If the trait is continuous with a Gaussian
distribution, e.g. from a Brownian motion (BM) or an Ornstein-Uhlenbeck (OU) process Hansen [1997], then the
likelihood at an ancestral species is a nice function of its state that can be concisely parametrized by quantities akin to
the posterior mean and variance conditional on descendant leaves. Felsenstein’s independent contrasts (IC) Felsenstein
[1985] also captures these partial posterior quantities and can be viewed as a special implementation of BP for likelihood
calculation.
BP is used ubiquitously for the analysis of discrete traits, such as for DNA substitution models (e.g. in
RAxML
Stamatakis
[2014],
IQ-TREE
Nguyen et al. [2015],
MrBayes
Ronquist and Huelsenbeck [2003]) or for discrete morphological
traits in comparative methods (e.g. in
phytools
Revell [2012],
BayesTraits
Pagel et al. [2004],
corHMM
Boyko and
Beaulieu [2021], Boyko et al. [2023],
RevBayes
H
¨
ohna et al. [2016]). For discrete traits, there is simply no feasible
alternative. On a tree with 20 taxa and 19 ancestral species, the naive calculation of the likelihood at a given DNA site
would require the calculation and summation of
419
or 274 billion likelihoods, one for each nucleotide assignment at
the 19 ancestral species. This calculation would need to be repeated for each site in the alignment, then repeated all
over during the search for a well-fitting phylogenetic tree.
2.2 Continuous traits on trees: the lazy way
For continuous traits under a Gaussian model (including the Brownian motion), BP is not used as ubiquitously because
a multivariate Gaussian distribution can be nicely captured by its mean and covariance matrix: the multivariate Gaussian
formula can serve as an alternative. For example, for one trait
Y
with ancestral state
µ
at the root of the phylogeny, the
phylogenetic covariance
Σ
between the taxa at the leaves can be obtained from the branch lengths in the tree. Under a
BM, the covariance
cov(Yi, Yj)
between taxa
i
and
j
is
Σij =σ2tij
where
tij
is the length between the root and their
most recent common ancestor. The likelihood of the observed traits at the
n
leaves can then be calculated using matrix
and vector multiplication techniques as
(2π)−n/2det |Σ|−1/2exp −1
2(Y−µ)⊤Σ−1(Y−µ).(1)
This alternative to BP has the disadvantage of requiring the inversion of the covariance matrix
Σ
, a task whose
computing time typically grows as
m3
for a matrix of size
m×m
. It also has the disadvantage that
Σ
needs to be
calculated and stored in memory in the first place. For multivariate observations of
p
traits on each of
n
taxa, the
covariance matrix has size
m=pn
so the typical calculation cost of
(1)
is then
O(p3n3)
, which can quickly become
very large. For example, with only 30 taxa and 10 traits,
Σ
is a
300 ×300
-matrix. Studies with large
p
and/or large
n
are now frequent, especially from geometric morphometric data with
p
over 100 typically (e.g. Hedrick [2023]) or with
expression data on
p > 1000
genes easily, that also require more complex models to account for variation (e.g. within
species, between organs, between batches) Dunn et al. [2013], Shafer [2019]. Studies with a large number
n
of taxa are
now frequent (e.g.
n > 5,000
in birds and mammals Jetz et al. [2012], Upham et al. [2019]) and virus phylogenies can
be massive (e.g.
n > 1000
and
p= 3
virulence traits in HIV Hassler et al. [2022a], or
n > 500,000
SARS-CoV-2
strains De Maio et al. [2023]).
In these cases with large data size
np
, the matrix-based alternative to BP is prone to numerical inaccuracy and numerical
instability in addition to the increased computational time, because it is hard to accurately invert a large matrix. Even
when the matrix is of moderate size, numerical inaccuracy can arise when the matrix is “ill-conditioned”. These
problems were identified under OU models on phylogenetic trees that have closely-related sister taxa, or under early-
burst (EB) models with strong morphological diversification early on during the group radiation, and much slowed-down
evolution later on Adams and Collyer [2017], Jhwueng and O’Meara [2020], Bartoszek et al. [2023].
For some simple models, the large
np ×np
covariance matrix can be decomposed as a Kronecker product of a
p×p
trait covariance and a
n×n
phylogenetic covariance. This decomposition can simplify the complexity of calculating
the likelihood. However, this decomposition is not available under many models, such as the multivariate Brownian
motion with shifts in the evolutionary rates (e.g. Caetano and Harmon [2019]) or the multivariate Ornstein-Uhlenbeck
model with non-scalar rate or selection matrices Bartoszek et al. [2012], Clavel et al. [2015].
4

Leveraging graphical model techniques to study evolution on phylogenetic networks
2.3 BP for continuous traits on trees
To bypass the complexity of matrix inversion, Felsenstein pioneered IC to test for phylogenetic correlation between
traits, assuming a BM model on a tree Felsenstein [1985]. Many authors then used BP approaches to handle Gaussian
models beyond the BM FitzJohn [2012], Freckleton [2012], Cybis et al. [2015], Goolsby et al. [2017]. Notably, Ho
and An
´
e [2014] describe a fast algorithm that can be used for non-Gaussian models as well. Most recently, Mitov
et al. [2020] highlighted that BP can be applied to a large class of Gaussian models: including the BM and the OU
process with shifts and variation of rates and selection regimes across branches. Software packages that use these fast
BP algorithms include
phylolm
Ho and An
´
e [2014],
Rphylopars
Goolsby et al. [2017],
BEAST
Hassler et al. [2023]
or the most recent versions of hOUwie Boyko et al. [2023] and mvSLOUCH Bartoszek et al. [2023].
All the methods cited above only use the first post-order tree traversal of BP to compute the likelihood. A second preorder
traversal allows, in the Gaussian case, for the computation of the distribution of all internal nodes conditionally on the
model and on the traits values at the tips. These distributions can then be used for, e.g., ancestral state reconstruction
Lartillot [2014], expectation-maximization algorithms for shift detection in the optimal values of an OU Bastide et al.
[2018a], or the computation of the gradient of the likelihood in the BM Zhang et al. [2021], Fisher et al. [2021] or
general Gaussian model Bastide et al. [2021]. Such BP techniques have also been used for taking gradients of the
likelihood with respect to branch lengths in sequence evolution models Ji et al. [2020] or for phylogenetic factor analysis
Tolkoff et al. [2018], Hassler et al. [2022b].
2.4 From trees to networks
So far, Felsenstein’s pruning algorithm and related BP approaches have been restricted to phylogenetic trees, mostly.
There is now ample evidence that reticulation is ubiquitous in all domains of life from biological processes such as
lateral gene transfer, hybridization, introgression and gene flow between populations. Networks are recognized to
be better than trees for representing the phylogenetic history of species and populations in many groups. Although
current studies using networks have few taxa, typically between 10-20 (e.g. Nielsen et al. [2023]), they tend to have
increasingly more tips as network inference methods become more scalable (e.g.
n= 39
languages in Neureiter et al.
[2022]). As viruses are known to be affected by recombination, we also expect future virus studies to use large network
phylogenies Ignatieva et al. [2022], so that BP will become essential for network studies too. In this work, we describe
approaches currently used for trait evolution on phylogenetic networks. We argue that the field of evolutionary biology
would benefit from applying BP approaches on networks more systematically. Transferring knowledge from the mature
and rich literature on BP would advance evolutionary biology research when phylogenetic networks are used.
2.5 Current network approaches for discrete traits
For discrete traits on general networks, very few approaches use BP techniques as far as we know. For DNA data for
example,
PhyLiNC
Allen-Savietta [2020] and
NetRAX
Lutteropp et al. [2022] extend the typical tree-based model to
general networks, assuming no incomplete lineage sorting. That is, each site is assumed to evolve along one of the
trees displayed in the network, chosen according to inheritance probabilities at reticulate edges.
PhyLiNC
assumes
independent (unlinked) sites.
NetRAX
assumes independent loci, which may have a single site each. Each locus may
have its own set of branch lengths and substitution model parameters. Both methods calculate the likelihood of a network
N
via extracting its displayed trees and then applying BP on each tree. Similarly, comparative methods for binary and
multi-state traits implemented in
PhyloNetworks
also extract displayed trees and then apply BP on each displayed
tree Karimi et al. [2020]. While these approaches use BP on each displayed tree, a network with
h
reticulations can
have up to 2hdisplayed trees. This leads to a computational bottleneck when the number of reticulations increases.
BP approaches have also been used for models with incomplete lineage sorting, modelled by the coalescent Kingman
[1982]. Notably,
SNAPP
models the evolution of unlinked biallelic markers along a species tree, accounting for
incomplete lineage sorting Bryant et al. [2012]. This method was recently made faster with
SNAPPER
Stoltz et al. [2020]
and extended to phylogenetic networks with
SnappNet
Rabier et al. [2021]. The coalescent process introduces the
challenge that each site may evolve along any tree, depending on past coalescent events.
SNAPP
introduced a way to
bypass the difficulties of handling coalescent histories and hence decrease computation time. After we describe BP for
general graphical models, we recast this innovation as BP on a graphical model formulation of the problem.
BP was also used to calculate the likelihood of the joint sample frequency spectrum (SFS). To account for incomplete
lineage sorting on a tree, Kamm et al. [2017] use the continuous-time Moran model to reduce computational complexity,
and assume that each site undergoes at most one mutation. In
momi2
, Kamm et al. [2020] extend the approach to
phylogenetic networks by assuming a pulse of admixture at reticulations. The associated graphical model is much
simpler than that required by SNAPP or SnappNet thanks to the assumption of no recurrent mutation.
5

Leveraging graphical model techniques to study evolution on phylogenetic networks
2.6 Current network approaches for continuous traits
Compared to the rich toolkit available for the analysis of continuous traits on trees, the toolkit for phylogenetic networks
is still limited.
PhyloNetworks
includes comparative methods on networks Sol
´
ıs-Lemus et al. [2017], implemented in
Julia
Bezanson et al. [2017]. These methods extend phylogenetic ANOVA to networks, for a continuous response trait
predicted by any number of continuous or categorical traits, with residual variation being phylogenetically correlated.
So far, the models available in
PhyloNetworks
include the BM, Pagel’s
λ
, possible within-species variation, and shifts
at reticulations to model transgressive evolution Bastide et al. [2018b], Teo et al. [2023]. However, all calculations are
based on working with the full covariance matrix, without BP.
TreeMix
Pickrell and Pritchard [2012],
ADMIXTOOLS
Patterson et al. [2012], Maier et al. [2023],
poolfstat
Gautier et al. [2022] and
AdmixtureBayes
Nielsen et al.
[2023] use allele frequency as a continuous trait. They model its evolution along a network, or admixture graph, using
a Gaussian model in which the evolutionary rate variance is affected by the ancestral allele frequency Soraggi and
Wiuf [2019], Lipson [2020]. Again, these methods work with the phylogenetic covariance matrix, rather than BP
approaches. They also consider subsets of up to 4 taxa at a time via
f2
,
f3
and
f4
statistics, which simplifies the
likelihood calculation. To identify selection and adaptation on a network,
PolyGraph
Racimo et al. [2018] and
GRoSS
Refoyo-Mart
´
ınez et al. [2019] assume a similar model and use the full covariance matrix. In summary, BP has yet to be
used for continuous trait evolution on networks.
3 Continuous trait evolution on a phylogenetic network
We now present phylogenetic models for the evolution of continuous traits, to which we apply BP later. We generalize
the framework in Mitov et al. [2020] and Bastide et al. [2021] from trees to networks, and we extend the network model
in Bastide et al. [2018b] from the BM to more general evolutionary models. We consider a multivariate
X
consisting of
p
continuous traits, and model their correlation over time. Our model ignores the potential effects of incomplete lineage
sorting on X, a reasonable assumption for highly polygenic traits.
3.1 Linear Gaussian models
Most random processes used to model continuous trait evolution on a phylogenetic tree are extensions of the BM to
capture processes such as evolutionary trends, adaptation, and variation in rates across lineages for example. In its most
general form, the linear Gaussian evolutionary model on a tree (referred to as the GLInv family in Mitov et al. [2020])
assumes that the trait Xvat node vhas the following distribution conditional on its parent pa(v)
Xv|Xpa(v)∼ N(qvXpa(v)+ωv,Vv)(2)
where the actualization matrix
qv
, the trend vector
ωv
and the covariance matrix
Vv
are appropriately sized and do
not depend on trait values
Xpa(v)
. When the tree is replaced by a network, a node
v
can have multiple parents
pa(v)
.
In this case, we can write
Xpa(v)
as the vector formed by stacking the elements of
{Xu|u∈pa(v)}
vertically, with
length equal to the number of traits times the number of parents of
v
. In the following, we show that
(2)
, already used
on trees, can easily be extended to networks, to describe both evolutionary models along one lineage and a merging rule
at reticulation events.
3.2 Evolutionary models along one lineage
For a tree node
v
with parent node
u
, we need to describe the evolutionary process along one lineage, graphically
modelled by the tree edge
e= (u, v)
. It is well known that a wide range of evolutionary models can fit in the general
form
(2)
Mitov et al. [2020], Bastide et al. [2021]. For instance, the BM with variance rate
Σ
(a variance-covariance
matrix for a multivariate trait) is described by
(2)
where
qv
is the
p×p
identity matrix
Ip
, there is no trend
ωv=0
,
and the variance is proportional to the edge length ℓ(e):Vv=ℓ(e)Σ.
Allowing for rate variation amounts to letting the variance rate vary across edges
Σ=Σ(e)
. For example, the Early
Burst (EB) model assumes that the variance rate at any given point in the phylogeny depends on the time
t
from the root
to that point, as:
Σ(t) = Σ0ebt .
For this
t
to be well-defined on a reticulate network, the network needs to be time-consistent (distinct paths from the
root to a node all share the same length). The rate
b
is a rate of variance decay if it is negative, to expected during
adaptive radiations, with a burst of variation near the root (hence Early Burst) before a slow-down of trait evolution
Harmon et al. [2010]. When
b > 0
, this model is called “accelerating rate” (AC) Blomberg et al. [2003]. Clavel and
6

Leveraging graphical model techniques to study evolution on phylogenetic networks
Morlon [2017] used a flexible extension of this model (on a tree), replacing
t
by one or more covariates that are known
functions of time, such as the average global temperature and other environmental variables:
Σ(t) = ˜σ(t, T1(t),··· , Tk(t)) .
Then, the variance accumulated along edge e= (u, v)is given by
Vv=Zt(v)
t(u)
Σ(t)dt .
In the particular case of the EB model, we get
Vv=Σ0ebt(u)(ebℓ(e)−1)/b .
Allowing for shifts in the trait value, perhaps due to jumps or cladogenesis, amounts to including ωv= 0 for some v.
Adaptive evolution is typically modelled by the OU process, which includes a parameter
Ae
for the strength of selection
along edge
e
. This selection strength is often assumed constant across edges, and is typically denoted as
α
for a
univariate trait. The OU process also includes a primary optimum value
θe
, which may vary across edges when we are
interested in detecting shifts in the adaptive regime across the phylogeny. Under the OU model, the trait evolves along
edge ewith random drift and a tendency towards θe:
dX(e)(t) = Ae(θe−X(e)(t))dt +RedB(t)
where
B
is a standard BM and the drift variance is
Σe=ReR⊤
e
. Then, conditional on the starting value at the start of
e
, the end value
Xv
is linear Gaussian as in
(2)
with actualization
qv=e−ℓ(e)Ae
, trend
ωv= (I−e−ℓ(e)Ae)θe
and
variance
Vv=Zℓ(e)
0
e−sAeΣee−sA⊤
eds =Se−e−ℓ(e)AeSee−ℓ(e)A⊤
e
where
Se
is the stationary variance matrix. These equations simplify greatly if
Ae
and
Σe
commute, such as if
Ae
is
scalar of the form αeIp, including when the process is univariate. In this case,
Vv= (1 −e−2αℓ(e))Σe/(2α).
Shifts in adaptive regimes can be modelled by shifts in any of the parameters θe,Aeor Σeacross edges.
Finally, variation within species, including measurement error, can be easily modelled by grafting one or more edges at
each species node, to model the fact that the measurement taken from an individual may differ from the true species
mean. The model for within-species variation, then, should also follow
(2)
by which an individual value is assumed
to be normally distributed with a mean that depends linearly on the species mean, and a variance independent of the
species mean – although this variance can vary across species. Most typically, observations from species
v
are modelled
using
q=Ip
,
ω=0
and some phenotypic variance to be estimated, that may or may not be tied to the evolutionary
variance parameter from the phylogenetic model across species. This additional observation layer can also be used for
factor analysis, where the unobserved latent trait evolving on the network has smaller dimension than the observed trait.
In that case, qis a rectangular, representing the loading matrix Tolkoff et al. [2018], Hassler et al. [2022b].
3.3 Evolutionary models at reticulations
For a continuous trait and a hybrid node
h
, Bastide et al. [2018b] and Pickrell and Pritchard [2012] assumed that
Xh
is
a weighted average of its immediate parents, using their state immediately before the reticulation event. Specifically,
if
h
has parent edges
e1, . . . , em
, and if we denote by
Xek
the state at the end of edge
ek
right before the reticulation
event (1≤k≤m), then the weighted-average model assumes that
Xh=X
ekparent of h
γ(ek)Xek.(3)
This model is a reasonable null model for polygenic traits, reflecting the typical observation that hybrid species show
intermediate phenotypes. In this model, the biological process underlying the reticulation event (such as gene flow
versus hybrid speciation) does not need to be known. Only the proportion of the genome inherited by each parent,
γ(ek)
, needs to be known. Compared to the evolutionary time scale of the phylogeny, the reticulation event is assumed
to be instantaneous.
To describe this process as a graphical model, we may add a degree-2 node at the end of each hybrid edge
e
to store
the value Xe, so as to separate the description of the evolutionary process along each edge from the description of the
7

Leveraging graphical model techniques to study evolution on phylogenetic networks
process at a reticulation event. With these extra degree-2 nodes, the weighted-average model
(3)
corresponds to the linear
Gaussian model
(2)
with no trend
ωh=0
, no variance
Vh=0
, and with actualization
qh= [γ(e1)Ip. . . γ(em)Ip]
made of scalar diagonal blocks.
Several extensions of this hybrid model can be considered. Bastide et al. [2018b] modelled transgressive evolution with
a shift
ωh=0
, for the hybrid population to differ from the weighted average of its immediate parents, even possibly
taking a value outside their range. Jhwueng and O’Meara [2015] considered transgressive shifts at each hybrid node as
random variables with a common variance, corresponding to a model with ωh=0but non-zero variance Vh.
More generally, we may consider models in which the hybrid value is any linear combination of its immediate parents
qvXpa(v)
as in
(2)
. A biologically relevant model could consider
qv
to be diagonal, with, on the diagonal, parental
weights γ(e, j)that may depend on the trait jinstead of being shared across all ptraits.
We may also consider both a fixed transgressive shift
ωh=0
and an additional hybrid variance
Vh
. For both of these
components to be identifiable in the typical case when we observe a unique realization of the trait evolution, the model
would need extra assumptions to induce sparsity. For example, we may assume that
Vh
is shared across all reticulations
and is given an informative prior, to capture small variations around the parental weighted average. We may also need a
sparse model on the set of
ωh
parameters, e.g. letting
ωh=0
only at a few candidate reticulations
h
, chosen based on
external domain knowledge.
For a continuous trait known to be controlled by a single gene, we may prefer a model similar to the discrete trait
model presented later in Example 2, by which
Xh
takes the value of one of its immediate parent
Xe
with probability
γ(e)
. This model would no longer be linear Gaussian, unless we condition on which parent is being inherited at each
reticulation. Such conditioning would reduce the phylogeny to one of its displayed tree. But it would require other
techniques to integrate over all parental assignments to each hybrid population, such as Markov Chain Monte Carlo or
Expectation Maximization.
3.4 Evolutionary models with interacting populations
Models have been proposed in which the evolution of
X(e)(t)
along one edge
e
depends on the state on other edges
existing at the same time
t
Drury et al. [2016], Manceau et al. [2017], Bartoszek et al. [2017], Duchen et al. [2020].
These models can describe “phenotype matching” that may arise from ecological interactions (mutualism, competition)
or demographic interactions (migration), in which traits across species converge to or diverge from one another. To
express this coevolution, we consider the set
E(t)
of edges contemporary to one another at time
t
and divide the
phylogeny into epochs: time intervals
[τi, τi+1]
during which the set
E(t)
of interacting lineages is constant, denoted
as
Ei
. Within each epoch
i
(i.e
t∈[τi, τi+1]
), the vector of all traits
(X(e)(t))e∈Ei
is modelled by a linear stochastic
differential equation. Since its mean is linear in and its variance independent of the starting value
(X(e)(τi))e∈Ei
, these
models are linear Gaussian Manceau et al. [2017], Bartoszek et al. [2017]. In fact, they can be expressed by
(2)
on a
supergraph of the original phylogeny, in which an edge
(u, v)
is added if
u
is at the start
τi
of some epoch
i
,
v
is at the
end
τi+1
, and if the mean of
Xv
conditional on all traits at time
τi
has a non-zero coefficient for
Xu
. The specific form
of
qv
,
ωv
and
Vv
in
(2)
depend on the specific interaction model, and may be more complex than the merging rule
(3)
.
4 A short review of graphical models and belief propagation
Implementing BP techniques on general networks is more complex than on trees. To explain why, we review here the
main ideas of graphical models and belief propagation for likelihood calculation.
4.1 Graphical models
A probabilistic graphical model is a graph representation of a probability distribution. Each node in the graph represents
a random variable, typically univariate but possibly multivariate. We focus here on graphical models with directed
edges. Edges represent dependencies between variables, where the direction is typically used to represent causation.
The graph expresses conditional independencies satisfied by the joint distribution of all the variables at all nodes in
the graph. Given the directional nature of evolution and inheritance, models for trait evolution on a phylogeny are
often readily formulated as directed graphical models. H
¨
ohna et al. [2014] demonstrate the utility of representing
phylogenetic models as graphical models for exposing assumptions, and for interpretation and implementation. They
present a range of examples common in evolutionary biology, with a focus on how graphical models facilitate greater
modularity and transparency. Here we focus on the computational gains that BP allows on graphical models.
A directed graphical model consists of a directed acyclic graph (DAG)
G
and a set of conditional distributions, one for
each node in
G
. At a node
v
with parent nodes
pa(v)
, the distribution of variable
Xv
conditional on its set of parent
8

Leveraging graphical model techniques to study evolution on phylogenetic networks
variables
Xpa(v)={Xu;u∈pa(v)}
is given by a factor
ϕv
, which is a function whose scope is the set of variables
from
v
and
pa(v)
. For each node
v
, the set formed by this node and its parents
{v} ∪ pa(v)
is called a node family. If
Vdenotes the vertex set of G, then the set of factors {ϕv, v ∈V}defines the joint density of the graphical model as
pθ(Xv;v∈V) = Y
v∈V
ϕv(Xv|Xu, θ;u∈pa(v)) (4)
where we add the possible dependence of factors on model parameters
θ
. This factor formulation implies that,
conditional on its parents,
Xv
is independent of any non-descendant node (e.g. “grandparents”) Koller and Friedman
[2009].
4.2 Phylogenetic examples of graphical models
Example 1 (BM on a tree).First consider the phylogenetic tree Tin Fig. 1a. The graphical model for the node states
of
T
under a BM, whose parameters
θ
are the trait evolutionary variance rate
σ2
, the ancestral state at the root
xρ
and
edge lengths
ℓi
, has the same topology as
T
. On a tree, each node family consists of a node
v
and its single parent, or
the root
ρ
by itself. The distribution
ϕρ
may be deterministic as when
xρ
is a fixed parameter of the model, or may be
given a prior distribution ϕρ.
T=G
(a)
xρ
x5x6
x1x2x3x4
U
(b)
xρ
x5, xρx6, xρ
x1, x5x2, x5x3, x6x4, x6
ℓ5ℓ6
ℓ1ℓ2ℓ3ℓ4x5x5x6x6
xρxρ
Figure 1: Example graphical model on a phylogenetic tree with factors defined by the BM. The joint distribution of
all variables at all nodes is given by the product of factors:
Qvϕv
where
ϕv
is the distribution of
xv
conditional on
its parent variable xpa(v):N(xpa(v), σ2ℓv)under the BM. (a) Phylogenetic tree T. (b) Clique tree Ufor the graphical
model. Its nodes are clusters of variables in
T
(ellipses). Each edge is labelled by a sepset (squares): a subset of
variables shared by adjacent clusters.
Example 2 (Discrete trait on a network).For a second example, we will consider a reticulate phylogeny. A rooted
phylogenetic network is a DAG with a single root, and taxon-labelled leaves (or tips). A node with at most one parent
is called a tree node and its incoming edge is a tree edge. A node with multiple parents is called a hybrid node, and
represents a population (or species more generally) with mixed ancestry. An edge
e= (u, h)
going into a hybrid node
h
is called a hybrid edge. It is assigned an inheritance probability
γ(e)>0
that represents the proportion of the genome
in
h
that was inherited from the parent population
u
(via edge
e
). Obviously, at each hybrid node
h
we must have
Pu∈pa(h)γ((u, h)) = 1
. The phylogenetic network
N
in Fig. 2a has one hybrid node
x5
whose genetic makeup comes
from x4with proportion 0.4and from x6with proportion 0.6.
N=G
(a)
xρ
x4x5x6
x1x2x3
U
(b)
x4, x6, xρ
x5, x4, x6
x1, x4x2, x5x3, x6
U∗
(c)
xρ
x4, xρx6, xρ
x5, x4, x6
γ= 0.4γ= 0.6
x4x5x6
x4, x6
x4x6
xρxρ
Figure 2: (a) Phylogenetic network
N
with hybrid edges shown in blue.
N
displays two trees, depending on which
hybrid edge is retained. One tree, with sister taxa 1 and 2, has probability
0.4
. The other tree, with sister taxa 2 and 3, is
displayed with probability
0.6
. The distribution of the hybrid node
x5
depends on both its parents, and induces a factor
cluster
{x4, x5, x6}
of size 3 in
U
and
U∗
. (b) Clique tree
U
for the graphical model. (c) Cluster graph
U∗
(leaf clusters
not shown) for the same graphical model in which
{x4, x6, xρ}
in
U
is replaced by smaller clusters
{x4, xρ}
,
{x6, xρ}
and {xρ}that induce a cycle.
9

Leveraging graphical model techniques to study evolution on phylogenetic networks
For a discrete trait
X
, the traditional model of evolution on a tree can be extended to a network
N
as follows. Along
each edge
e
,
X
evolves according to a Markov process with some transition rate matrix
Q
for an amount of time
ℓ(e)
that depends on the edge. At a tree node, the state of
X
at the end of its parent edge is passed as the starting value to
each daughter lineage, as in the traditional tree model. At reticulations, we follow previous authors to model the value
xh
at a hybrid node
h
Karimi et al. [2020], Allen-Savietta [2020], Lutteropp et al. [2022]. Let
xe
denote the state at the
end of edge
e
. If
h
has
m
parent edges
e1,··· , em
, then
xh
is assumed to take value
xek
with probability
γ(ek)
. This
model reflects the idea that the trait is controlled by unknown genes, but the proportion of genes inherited from each
parent is known. Incomplete lineage sorting, which can lead to hemiplasy for a trait Avise and Robinson [2008], is
unaccounted for. Similar to Example 1, the graphical model uses the topology of the network N.
To describe the factors of this graphical model and simplify notations, consider the case when
X
is binary with states
0 and 1. For a tree node
v
, the factor
ϕv
can be represented by the
2×2
matrix
exp(ℓ(e)Q)
, where
e
is the parent
edge of
v
. For a hybrid node
h
with
m
parents
p1,··· , pm
and edges
ek= (pk, h)
with
γ(ek) = γk
, the factor
ϕh
has scope
(Xh, Xp1,··· , Xpm)
, and can be described by a
2×2m
matrix to store the conditional probabilities
P(Xh=j|Xp1=i1,··· , Xpm=im)
. This is a
2×4
matrix in the typical case when
h
is admixed from
m= 2
parental populations. With
m= 2
and with parental values
(Xp1, Xp2)
arranged ordered
((0,0),(0,1),(1,0),(1,1))
,
then
ϕh=1γ1γ20
0γ2γ11.
T
(a)
nρ,rρ
n3,r3
n1,r1n2,r2
G
(b)
nρrρ
n3r3
n3
r3
n1r1r2n2
n1r1r2n2
U
(c)
nρ,rρ, n3,r3
n3,r3, n3,r3
n1,r1, n2,r2, n3,r3
n1,r1, n1,r1n2,r2, n2,r2
n3,r3
n1,r1n2,r2
n1,r1n2,r2
n3,r3
n3,r3
Figure 3: (a) Phylogenetic species tree
T
, used to generate gene genealogies for
n1
and
n2
individuals sampled from
species 1 and 2 respectively. (b) Graph
G
for the graphical model associated with the evolution of a binary trait on a
gene tree drawn from the multispecies coalescent model.
G
is a DAG with 2 sources (roots)
n1
and
n2
, and 2 sinks
(leaves)
r1
and
r2
. The model restricted to variables
ne
and
ne
(in black) can be described by the subgraph
Gn
whose
nodes and edges are in black. It is a tree similar to
T
but with reversed edge directions. (c) Clique tree
U
for
G
. Note
that the 6-variable clique is overparametrized because
n1+n2=n3
and
r1+r2=r3
, but reflects the symmetry of the
model.
Example 3 (Binary trait with ILS on a tree).Our final example is a case that accounts for incomplete lineage sorting
(ILS), when the graph
G
for the graphical model is constructed from but not identical to the phylogeny. Consider the
species tree in Fig. 3a, a sample of one or more individuals sampled from each species (1 and 2), and a gene tree (or
genealogy) generated according to the multispecies coalescent model along
T
Kingman [1982], Rannala and Yang
[2003]. Finally, consider a binary trait evolving along this gene tree, with states (or alleles) “black” and “red” to re-use
terminology by Bryant et al. [2012]. The observations from this model are the number of individuals
ri
with the red
allele among the
ni
individuals sampled from each species
i
. Bryant et al. [2012] discovered conditional independencies
in this model by considering and conditioning on the total number (
n
) and number of red alleles (
r
) ancestral to the
sampled individuals at the beginning of each edge
e
:
ne
and
re
; and at the end of each edge
e
:
ne
and
re
. Here, we
formulate this evolutionary model as a graphical model. Its graph
G
is different from the original phylogenetic tree, as
illustrated in Fig. 3b.
If we only consider the ancestral number of individuals
n
, then the graph
Gn
for the associated graphical model is
as follows, thanks to the description of the coalescent model going back in time. For each edge
e
in
T
, an edge is
created in
Gn
but with the reversed direction (black subgraph in Fig. 3). On this edge, the coalescent edge factor
ϕne=P(ne|ne)
was derived by Tavar
´
e [1984] and is given in [Bryant et al., 2012, eq. (6)]. Each internal node
v
in
T
is triplicated in
Gn
to hold the variables
ne
,
nc1
and
nc2
, where
e
denotes the parent edge of
v
and
c1, c2
denote its
child edges (assuming that
v
has only 2 children, without loss of generality). These nodes are then connected in
Gn
by
10

Leveraging graphical model techniques to study evolution on phylogenetic networks
edges from each
nci
to
ne
. The speciation factor
ϕne=
1
{nc1+nc2}(ne)
expresses the relationship
ne=nc1+nc2
.
Overall, Gnis a tree with a single sink (leaf), multiple sources (roots), and data at the roots.
To calculate the likelihood of the data, we add the number of red alleles
r
ancestral to the sampled individuals.
The full graph
G
(Fig. 3b) contains
Gn
, with extra nodes for the
r
variables, and extra edges to model the process
along edges and at speciations. The node family for
re
includes
re
and both
ne
and
ne
. The mutation edge factor
ϕre=P(re|re, ne, ne)
was derived by Griffiths and Tavar
´
e [1994] using both the coalescent and mutation processes,
and is given in [Bryant et al., 2012, eq. (16)]. For edge
e
in
T
with child edges
c1
and
c2
in
T
, the speciation factors for
red alleles
ϕrc1=P(rc1|nc1, ne, re)
and
ϕrc2=
1
{re−rc1}(rc2)
describe a hypergeometric distribution where
nc1
individuals, rc1of which are red, are sampled from a pool of neindividuals, reof which are red, and rc2=re−rc1.
Given this graphical model description, the likelihood calculation described in Bryant et al. [2012] corresponds to BP
along graph G, as we will illustrate later.
This framework can be extended to the case when the phylogeny is reticulate, with additional edges in
G
, and
hybridization factors to model the process at hybrid nodes for the
n
and
r
variables, illustrated on an example in SM
section A. The likelihood calculations used in
SnappNet
and described in Rabier et al. [2021] correspond to BP along
this graph G.
4.3 Belief Propagation
BP is a framework for efficiently computing various integrals of the factored density
pθ
by grouping nodes and their
associated variables into clusters and integrating them out according to rules along a clique tree (also known by junction
tree, join tree, or tree decomposition) or along a cluster graph, more generally.
4.3.1 Cluster graphs and Clique trees
Definition 1 (cluster graph and clique tree).Let
Φ = {ϕv, v ∈V}
be the factors of a graphical model on graph
G
and
let
U= (V,E)
be an undirected graph whose nodes
V
, called clusters, consists of sets of variables in the scope of
Φ
.
U
is a cluster graph for Φif it satisfies the following properties:
1.
(family-preserving) There exists a map
α: Φ → V
such that for each factor
ϕv
, its scope (node family for
node vin the graphical model) is a subset of the cluster α(ϕv).
2.
(edge-labeled) Each edge
{Ci,Cj}
in
E
is labelled with a non-empty sepset
Si,j
(“separating set”) such that
Si,j ⊆ Ci∩ Cj.
3.
(running intersection) For each variable
x
in the scope of
Φ
,
Ex⊆ E
, the set of edges with
x
in their sepsets
forms a tree that spans Vx⊆ V, the set of clusters that contain x.
If
U
is acyclic, then
U
is called a clique tree and we refer to its nodes as cliques. In this case, properties 2 and 3 imply
that Si,j =Ci∩ Cj.
A clique tree
U
is shown in Fig. 1b for the BM model from Example 1, on the tree
T
in Fig. 1a. To check the running
intersection property for
x5
, for example, we extract the graph defined by edges with
x5
in their sepsets (squares).
There are 2 such edges. They induce a subtree of
U
that connects all 3 clusters (ellipses) containing
x5
, as desired.
More generally, when the graphical model is defined on a tree
T
, a corresponding clique tree
U
is easily constructed,
where cliques in
U
correspond to edges in
T
, and edges in
U
correspond to nodes in
T
. Multiple clique trees can be
constructed for a given graphical model. In this example, the clique
{xρ}
(shown at the top) could be suppressed,
because it is a subset of adjacent cliques.
For the network
N
in Fig. 2a and the evolution of a discrete trait in Example 2, one possible clique tree
U
is shown
in Fig. 2b. Note that
x5, x4
and
x6
have to appear together in at least one of the clusters for the clique tree to be
family-preserving (property 1), because
x4
and
x6
are partners with a common child
x5
whose distribution depends on
both of their states.
We first focus on clique trees, which provide a structure for the exact likelihood calculation. In section 5 we discuss the
advantages of cluster graphs, to approximate the likelihood at a lower computational cost.
4.3.2 Evidence
To calculate the likelihood of the data, or the marginal distribution of the traits at some node conditional on the data, we
inject evidence into the model, in one of two equivalent ways. For each observed value
xv,t
of the
tth
trait
xv,t
at node
11

Leveraging graphical model techniques to study evolution on phylogenetic networks
v
, we add to the model the indicator function
1
{xv,t}(xv ,t)
as an additional factor. Equivalently, we can plug in the
observed value
xv,t
in place of the variable
xv,t
in all factors where
xv,t
appears, and then drop
xv,t
from the scope
of all these factors. This second approach is more tractable than the first to avoid the degenerate zero-variance Dirac
distribution. But it requires careful bookkeeping of the scope and of re-parametrization of each factor with missing data,
when some traits but not all are observed at some nodes. Below, we assume that the factors and their scopes have been
modified to absorb evidence from the data.
4.3.3 Belief update message passing
There are multiple equivalent algorithms to perform BP. We focus here on the belief update algorithm. It assigns a
belief to each cluster and to each sepset in the cluster graph. After running the algorithm, each belief should provide the
marginal probability of the variables in its scope and of the observed data, with all other variables integrated out as
desired to calculate the likelihood. The belief
βi
of cluster
Ci
is initialized as the product of all factors assigned to that
cluster:
ψi=Y
ϕ;α(ϕ)=Ci
ϕfor cluster Ci(5)
The belief
µi,j
of an edge between cluster
i
and
j
is initialized to the constant function 1. These beliefs are then updated
iteratively by passing messages. Passing a message from
Ci
to
Cj
along an edge with sepset
Si,j
corresponds to passing
information about the marginal distribution of the variables in
Si,j
as shown in Algorithm 1. If
U
is a clique tree, then
Algorithm 1 Belief propagation: message passing along an edge from Cito Cjwith sepset Si,j .
1:
compute the message
˜µi→j=RCi\Si,j βid(Ci\Si,j )
, that is, the marginal probability of
Si,j
based on belief
βi
, by
integrating all other variables in Ci,
2: update the cluster belief about Cj:βj←βj˜µi→j/µi,j ,
3: update the edge belief about Si,j :µi,j ←˜µi→j.
all beliefs converge to the true marginal probability of their variables and of the observed data, after traversing
U
only
twice: once to pass messages from leaf cliques towards some root clique, and then back from the root clique to the
leaf cliques. If our goal is to calculate the likelihood, then one traversal is sufficient. Once the root clique has received
messages from all its neighboring cliques, we can marginalize over all its variables (similar to step 1) to obtain the
probability of the observed data only, which is the likelihood. The second traversal is necessary to obtain the marginal
probability of all variables, such as if one is interested in the posterior distribution of ancestral states conditional on the
observed data.
Some equivalent formulations of BP only store sepset messages, and avoid storing cluster beliefs. This strategy requires
less memory but more computing time if Uis traversed multiple times.
Example 4 (link to IC).Continuing on Example 1 on the tree in Fig. 1, the conditional distribution of
xv
at a non-root
node
v
corresponds to a factor
ϕv
for the BM model along edge
(pa(v), v)
in
T
. This factor is assigned to clique
Cv={pa(v), v}
in
U
to initialize the belief
βv
of
Cv
. If
v
if a leaf in
T
, then
βv
is further multiplied by the indicator
function at the value
xv
observed at
v
, such that the belief of clique
Cv
can be expressed as a function of the leaf’s parent
state only:
ϕv(xpa(v)) = P(xv|xpa(v))
. The prior distribution
ϕ(xρ)
at the root
ρ
of
T
(which can be an indicator
function if the root value is fixed as a model parameter) can be assigned to any clique containing
ρ
. In Fig. 1,
U
includes
a clique
Cρ={xρ}
drawn at the top, to which we assign the root prior
ϕρ(xρ)
and which we will use as the root of
U
. Since
U
is a clique tree, BP converges after traversing
U
twice: from the tips to
Cρ
and then back to the tips. IC
Felsenstein [1973, 1985] implements the first “rootwards” traversal of BP. For example, the belief of clique
{x5, xρ}
after receiving messages (steps 1-3) from both of its daughter cliques is the function
β5(x5, xρ) = exp −(xρ−x5)2
2ℓ5−(x5−x∗
5)2
2v∗
5
+g∗
5
where
x∗
5=ℓ2x1+ℓ1x2
ℓ1+ℓ2
, v∗
5=ℓ1ℓ2
ℓ1+ℓ2
,and g∗
5=−(x2−x1)2
2(ℓ1+ℓ2)−log((2π)3/2ℓ1ℓ2ℓ5)
are quantities calculated for IC:
x∗
5
corresponds to the estimated ancestral state at node 5,
v∗
5
corresponds to the extra
length added to
ℓ5
when pruning the daughters of node 5, and
g∗
5
captures the contrast
(x2−x1)/√ℓ1+ℓ2
below node
5. At this stage of BP,
β5(x5, xρ)
can be interpreted as
P(x1,x2, x5|xρ)
such that the message
˜µ5→ρ(xρ)
sent from
{x5, xρ}
to the root clique
Cρ
is the partial likelihood
P(x1,x2|xρ)
after
x5
is integrated out. The first pass is complete
when
Cρ
has received messages from all its neighbors. Its final belief is then
βρ(xρ) = P(x1,··· ,x4|xρ)ϕρ(xρ)
. If
xρ
is a fixed model parameter, then this is the likelihood. Otherwise, we get the likelihood by integrating out
xρ
in
βρ(xρ)
.
12

Leveraging graphical model techniques to study evolution on phylogenetic networks
In Example 2 on a network (Fig. 2), we label the cliques in
U
as follows:
Cv={xv, xpa(v)}
for leaves
v= 1,2,3
,
C5={x5, x4, x6}
for hybrid node
v= 5
and its parents, and
Cρ={x4, x6, xρ}
. To initialize beliefs, we assign
ϕv
to
Cv
for
v= 1,2,3,5
, and
ϕ4
,
ϕ6
are both assigned to
Cρ
. Unlike in Example 1 on a tree, a clique may correspond
to more than a single edge in
T
. This is expected at a hybrid node
h
, because the factor describing its conditional
distribution needs to contain
h
and both of its parents. But for
U
to be a clique tree, the root clique
Cρ
also has to
contain the factors from 2 edges in
T
. Also, unlike for trees, sepsets may contain more than a single node. Here, the
two large cliques are separated by
{x4, x6}
so they will send messages
˜µ(x4, x6)
about the joint distribution of these
two variables. In this binary trait setting, these messages and sepset belief can be stored as
2×2
arrays, and the 3-node
cliques beliefs can be stored as arrays of
23
values. As they involve more variables than when
G
is a tree (in which case
BP would store only 2 values at each sepset), storing and updating them requires more computating time and memory.
More generally, we see that the computational complexity of BP scales with the size of the cliques and sepsets. This
complexity may become prohibitive on a more complex phylogenetic network, even for a simple binary trait without
ILS, if the size of the largest cluster in Uis too large —a topic that we explore later.
Example 3 illustrates the fact that beliefs cannot always be interpreted as partial (or full) likelihoods at every step
of BP, unlike in Examples 1 and 2. For example, consider the first iteration of BP, with the tip clique
C1
containing
(n1, r1)
(Fig. 3) sending a message to its large neighbor clique. The belief of
C1
is initialized with the factors
ϕn1
and
ϕr1
, which are the probabilities of
n1
and of
r1
conditional on their parents in graph
G
. From fixing
(n1, r1)
to their
observed values (n1,r1), the message sent by C1in step 1 is
˜µ(n1, r1) = P(n1|n1)P(r1|r1, n1,n1).
This message is the quantity denoted by
FT(n, r)
in Bryant et al. [2012]. It is not a partial likelihood, because it is not
the likelihood of some partial subset of the data conditional on some ancestral values in the phylogeny. Intuitively, this
is because nodes with data below
n1
include both
r1
and
r2
, yet
C1
does not include
r2
. Information about
r2
will be
passed to the root of
U
separately. More generally, during the first traversal of
U
, each sepset belief corresponds to an
F
value in Bryant et al. [2012]:
FT
for sepsets at the top of a branch
(ne, re)
, and
FB
for sepsets at the bottom of a
branch
(ne, re)
. The beauty of BP on a clique tree is that beliefs are guaranteed to converge to the likelihood of the full
data, conditional on the state of the clique variables. After messages are passed down from the root to
C1
, the updated
belief of C1will indeed be the likelihood of the full data conditional on n1and r1.
4.3.4 Clique tree construction
For a given graphical model on
G
, there are many possible clique trees and cluster graphs. For running BP, it is
advantageous to have small clusters and small sepsets. Indeed, clusters and sepsets with fewer variables require less
memory to store beliefs, and less computing time to run steps 1 (integration) and 2 (belief update). Ideally, we would
like to find the best clique tree: whose largest clique is of the smallest size possible. For a general graph
G
, finding this
best clique tree is hard but good heuristics exist Koller and Friedman [2009].
The first step is to create the moralized graph
Gm
from
G
. This is done by connecting all nodes that share a common
child, and then undirecting all edges. We can then triangulate
Gm
, that is, build a new graph
H
by adding edges to
Gm
such that
H
is chordal (any cycle includes a chord). This is the hard step, if one wants to find a triangulation with
the smallest maximum clique size. An efficient heuristic is the greedy minimum-fill heuristic Rose [1972], Fishelson
and Geiger [2003]. The cliques in
U
are then taken as the maximal cliques in
H
Blair and Peyton [1993]. Finally, the
edges in
U
are formed such that
U
becomes a tree and such that the sum of the sepset sizes is maximum, by finding
a maximum spanning tree using Kruskal’s algorithm or Prim’s algorithm Cormen et al. [2009]. All these steps have
polynomial complexity.
4.4 BP for Gaussian models
Before discussing BP on cluster graphs that are not clique trees, we focus on BP updates for the evolutionary models
presented in section 3. On a phylogenetic network
N
, the joint distribution of all present and ancestral species
(Xv)v∈N
is multivariate Gaussian precisely when it comes from a graphical model on
N
whose factors
ϕv
are linear Gaussian
Koller and Friedman [2009]. The factor at node
v
is linear Gaussian if, conditional on its parents,
Xv
is Gaussian with
a mean that is linear in the parental values and a variance independent of parental values, hence the term
GLInv
used by
Mitov et al. [2020]. In other words, for the joint process to be Gaussian, each factor
ϕv(xv|xpa(v))
should be of the
form (2).
Such models have been called Gaussian Bayesian networks or graphical Gaussian networks, and are special cases of
Gaussian processes (on a graph). These Gaussian models are convenient for BP because linear Gaussian factors have a
13

Leveraging graphical model techniques to study evolution on phylogenetic networks
convenient parametrization that allows for a compact representation of beliefs and belief update operations. Namely,
the factor giving the conditional distribution
ϕv(xv|xpa(v))
from
(2)
can be expressed in a canonical form as the
exponential of a quadratic form:
C(x;K, h, g) = exp −1
2x⊤Kx+h⊤x+g.(6)
For example, if we think of
ϕv(xv|xpa(v))
as a function of
xv
primarily, we may use the parametrization
C(xv;K, h, g)
with
K=V−1
v, h =V−1
vqvxpa(v)+ωv,and g=−1
2log |2πVv|+∥qvxpa(v)+ωv∥2
V−1
v
where ∥y∥2
Mdenotes y⊤My. We can also express ϕvas a canonical form over its full scope
ϕv(xv|xpa(v)) = C xv
xpa(v);Kv, hv, gv
with
Kv=V−1
v−V−1
vqv
−q⊤
vV−1
vq⊤
vV−1
vqv=I
−q⊤
vV−1
v[I−qv], hv=V−1
vωv
−q⊤
vV−1
vωv, gv=−1
2(log |2πVv|+∥ωv∥V−1
v).
(7)
If
v
is a leaf with fully observed data, then we need to plug-in the data
xv
into
ϕv
and consider this factor as a function
of xpa(v)only. We can express ϕv(xv|xpa(v))as the canonical form C(xpa(v);K, h, g )with
K=q⊤
vV−1
vqv, h =q⊤
vV−1
v(xv−ωv),and g=−1
2log |2πVv|+∥xv−ωv∥2
V−1
v.
If data are partially observed at leaf
v
, the same principle applies. We can plug-in the observed traits into
ϕv
and express
ϕv
as a canonical form over its reduced scope:
xpa(v)
and any unobserved
xv,t
. Some quadratic terms captured by
Kv
on the full scope become linear or constant terms after plugging-in the data, and some linear terms captured by
hv
on
the full scope become constant terms in the canonical form on the reduced scope.
An important property of this canonical form is its closure under the belief update operations: marginalization (step 1)
and factor product (step 2). Indeed, the product of two canonical forms with the same scope satisfies:
C(x;K1, h1, g1)C(x;K2, h2, g2) = C(x;K1+K2, h1+h2, g1+g2).
Now consider marginalizing a factor
C(x;K, h, g)
to a subvector
x∗
of
x
, by integrating out the elements
x\x∗
of
x
.
let
KS
and
KI
be the submatrices of
K
that correspond to
x∗
(Scope of marginal or Sepset) and
x\x∗
(variables to be
Integrated out), and let KS,I=K⊤
I,Sbe the cross-terms. If KIis invertible, then:
ZCx\x∗(x;K, h, g)d(x\x∗) = C(x∗;K∗, h∗, g∗)
where
K∗=KS−KS,IK−1
IKI,S
,
h∗=hS−KS,IK−1
IhI
with
hS
and
hI
defined as the subvector of
h
corresponding
to x∗and x\x∗respectively, and g∗=g+ (log |2πK−1
I|+∥hI∥K−1
I)/2.
If the factors of a Gaussian network are non-deterministic, then each belief can be parametrized by its canonical form,
and the above equations can be applied to update the cluster and sepset beliefs for BP (Algorithm 1). For cluster
Ci
, let
(Ki, hi, gi)
parametrize its belief
βi
. For sepset
Si,j
, let
(Ki,j , hi,j , gi,j )
parametrize its belief
µi,j
. Also, for step 1 of
BP, let
(Ki→j, hi→j, gi→j)
parametrize the message
˜µi→j
sent from
Ci
to
Cj
. Then BP updates can be expressed as
shown below.
In step 1,
KS
and
KI
are the submatrices of
Ki
that correspond to
Si,j
and
Ci\Si,j
. Similarly,
hS
and
hI
are subvectors
of
hi
. In step 2,
ext(K˜µ−Ki,j )
extends
K˜µ−Ki,j
to the same scope as
Kj
by padding it with zero-rows and
zero-columns for
Cj\ Si,j
. Similarly,
ext(hi→j−hi,j )
extends
hi→j−hi,j
to scope
Cj
with
0
entries on rows for
Cj\ Si,j .
If the phylogeny is a tree, performing these updates from the tips to the root corresponds to the recursive equations (9),
(10) and (11) of Mitov et al. [2020], and to the propagation formulas (A.3)-(A.8) of Bastide et al. [2021], who both
considered the general linear Gaussian model (2).
At any point, a belief
C(x;K, h, g)
gives a local estimate of the conditional mean (
K−1h
) and conditional variance
(
K−1
) of trait
X
given data
Y
, for
K≻0
. An exact belief, such that
C(x;K, h, g)∝pθ(x|Y)
, gives exact
conditional estimates, that is: E(X|Y) = K−1hand var(X|Y) = K−1.
14

Leveraging graphical model techniques to study evolution on phylogenetic networks
Algorithm 2 Gaussian belief propagation: from Cito Cjwith sepset Si,j .
1: compute message ˜µi→j:




Ki→j=KS−KS,IK−1
IKI,S
hi→j=hS−KS,IK−1
IhI
gi→j=gi+ (log |2πK−1
I|+∥hI∥K−1
I)/2
2: update the cluster belief βjabout Cj:


Kj←Kj+ext(Ki→j−Ki,j)
hj←hj+ext(hi→j−hi,j)
gj←gj+gi→j−gi,j
3: update the edge belief µi,j about Si,j :


Ki,j ←Ki→j
hi,j ←hi→j
gi,j ←gi→j
5 Scalable approximate inference with loopy BP
The previous examples focused on clique trees and the exact calculation of the likelihood. We now turn to the use of
cluster graphs with cycles, or loopy cluster graphs, such as in Fig. 2(c) or Fig. 4(c-d). BP on a loopy cluster graph,
abbreviated as loopy BP, can approximate the likelihood and posterior distributions of ancestral values, and can be
more computationally efficient than BP on a clique tree.
(a) (b)
9,8,10,12 7,9,8,12
3,9,11 9,10,11,12 7,6,8,12 7,5,6,8
1,3,9 2,4,6 4,7,5,6
(c)
9,10,11 10,11,12 8,10,12 6,8,12
3,9,11 9,8 7,8,92,4,6
1,3,97,5,84,7,5
(d)
11,12 6,8,12 2,4,6
3,11 10,11 8,10 5,84,7,5
1,3,99,10 7,9
Figure 4: (a) Admixture graph
N
from [Lazaridis et al., 2014, Fig. 3] with
h= 4
reticulations (hybrid edges are
coloured).
N
has one non-trivial biconnected component (blob)
B
, induced by all its internal nodes except for the root.
B
contains all 4 reticulations so
N
has level
ℓ= 4
. (b)-(d) Various cluster graphs for the moralized blob
Bm
: (b) clique
tree, (c) join-graph structuring with the maximum cluster size set to 3, (d) LTRIP using the set of node families in
B
.
Here sepsets (not shown) are the intersection of their incident clusters, and are small with 1 node only in (c) and (d).
Purple boxes and edges: clusters and sepsets that contain node 8. Red text: hybrid families.
5.1 Calibration
Updating beliefs on a loopy cluster graph uses Algorithm 1 in the same way as on a clique tree. A cluster graph is said
to be calibrated when its normalized beliefs have converged (i.e. are unchanged by Algorithm 1 along any edge). For
calibration, neighboring clusters
Ci
and
Cj
must have beliefs that are marginally consistent over the variables in their
sepset Si,j :Zβid(Ci\Si,j ) = ˜µi→j∝µi,j ∝˜µj→i=Zβjd(Cj\Si,j ).
On a clique tree, calibration can be guaranteed at the end of a finite sequence of messages passed. Clique and sepset
beliefs are then proportional to the posterior distribution over their variables, and can be integrated to compute the
15

Leveraging graphical model techniques to study evolution on phylogenetic networks
common normalization constant
κ=κi=RβidCi=κj,k =Rµj,kdSj,k 
, which equals the likelihood. For loopy
BP, calibration is not guaranteed. If it is attained, then we can similarly view cluster and sepset beliefs as unnormalized
approximations of the posterior distribution over their variables, though the
κi
s and
κj,k
s may differ, grow unboundedly,
and generally do not equal or estimate the likelihood. Gaussian models enjoy the remarkable property that, if calibration
can be attained on a cluster graph, then the approximate posterior means (ancestral values) are guaranteed to be exact.
In contrast, the posterior variances are generally inexact, and are typically underestimated Weiss and Freeman [1999],
Wainwright et al. [2003], Malioutov et al. [2006], although we found them overestimated in our phylogenetic examples
below (Fig. 7).
Successful calibration depends on various aspects such as the features of the loops in the cluster graph, the factors in the
model, and the scheduling of messages. For beliefs to converge, a proper message schedule requires that a message is
passed along every sepset, in each direction, infinitely often (until stopping criteria are met) Malioutov et al. [2006].
Multiple scheduling schemes have been devised to help reach calibration more often and more accurately. These can
be data-independent (e.g. choosing a list of trees nested in the cluster graph that together cover all clusters and edges,
then iteratively traversing each tree in both directions Wainwright et al. [2003]) or adaptive (e.g. prioritizing messages
between clusters that are further from calibration Elidan et al. [2006], Sutton and McCallum [2007], Knoll et al. [2015],
Aksenov et al. [2020]).
5.2 Likelihood approximation
To approximate the log-likelihood
LL(θ) = log Rpθ(x)dx
from calibrated beliefs on cluster graph
U∗= (V∗,E∗)
,
denoted together as
q={βi, µi,j ;Ci∈ V∗,{Ci,Cj}∈E∗}
, we can use the factored energy functional Koller and
Friedman [2009]:
˜
F(pθ, q) = X
Ci∈V∗
Eβi(log ψi) + X
Ci∈V∗
H(βi)−X
{Ci,Cj}∈E∗
H(µi,j ).(8)
Recall that
ψi
is the product of factors
ϕv
assigned to cluster
Ci
. Here
Eβi
denotes the expectation with respect to
βi
normalized to a probability distribution.
H(βi)
and
H(µi,j )
denote the entropy of the distributions defined by
normalizing
βi
and
µi,j
respectively.
˜
F(pθ, q)
has the advantage of involving local integrals that can be calculated
easily: each over the scope of a single cluster or sepset. The justification for
˜
F(pθ, q)
comes from two approximations.
First, following the expectation-maximization (EM) decomposition,
LL(θ)
can be approximated by the evidence lower
bound (ELBO) used for variational inference Ranganath et al. [2014]. For any distribution
q
over the full set of variables,
which are here the unobserved (latent) variables after absorbing evidence from the data, we have
LL(θ)≥ELBO(pθ, q)=Eq(log pθ) + H(q).
The gap
LL(θ)−ELBO(pθ, q)
is the Kullback-Leibler divergence between
q
, and
pθ
normalized to the distribution of
the unobserved variables conditional on the observed data. The first approximation comes from minimizing this gap over
a class of distributions
q
that does not necessarily include the true conditional distribution. The second approximation
comes from pretending that for a given distribution qwith a belief factorization
q∝QCi∈V∗βi
Q{Ci,Cj}∈E∗µi,j
,
its marginal over a given cluster (or a given sepset) is equal to the normalized belief of that cluster (or sepset),
simplifying
Eq(log ψi)
to
Eβi(log ψi)
and simplifying
Eq(−log βi)
to
H(βi)
. This simplification leads to the more
tractable ˜
F(pθ, q), in which each integral is of lower dimension, within the scope of a single cluster or sepset.
5.3 Scalability versus accuracy: choice of cluster graph complexity
5.3.1 Scalability, treewidth and phylogenetic network complexity
At the cost of exactness, loopy cluster graphs can offer greater computational scalability than clique trees because they
allow for smaller cluster sizes, which reduces the complexity associated with belief updates. For example, consider
a Gaussian model for
p
traits:
dim(xv) = p
at all nodes
v
in the network. For a clique tree
U
with
m
cliques and
maximum clique size
k
, passing a message between neighbor cliques has complexity
O(p3k3)
and calibrating
U
has
complexity
O(mp3k3)
. Now consider a cluster graph
U∗
with
m∗
clusters,
O(m∗)
edges, and maximum cluster size
k∗< k
. Then passing a message between neighbor cliques of
U∗
has complexity
O(p3k∗3)
so it is faster than on
U
.
But calibrating
U∗
now requires more belief updates because each edge needs to be traversed more than twice. If each
edge is traversed in both directions
b
times to reach convergence, then calibrating
U∗
has complexity
O(bm∗p3k∗3)
. So
16

Leveraging graphical model techniques to study evolution on phylogenetic networks
N2u1
v1u2
v2u3
a b
N1˜w
w u3
u1u2b
c v1d
v2
a
u1and u2are adjacent
u3is a descendant of u2
Figure 5: Two binary networks with a hybrid ladder and
h=ℓ= 2
.
N1
satisfies (A2) of Proposition 1 and
Nm
1
has
treewidth
t= 3
.
N2
does not meet (A2) (see red/purple annotations) and
Nm
2
has treewidth
t= 2
. Stacking more
hybrid ladders in the same way above aand bincreases hand ℓbut leaves Nm
2outerplanar, keeping t= 2.
if
U∗
has smaller clusters than
U
and if
(k/k∗)3≫bm∗/m
, then loopy BP on
U∗
runs faster than BP on
U
. Loopy BP
could be particularly advantageous for complex networks whose clique trees have large clusters.
Cluster graph construction determines the balance between scalability and approximation quality. At one end of the
spectrum, the most scalable and least accurate are the factor graphs, also known as Bethe cluster graphs Yedidia et al.
[2005]. A factor graph has one cluster per factor
ϕv
and one cluster per variable, and so has the smallest possible
maximum clique size
k∗
and each sepset reduced to a single variable. Various algorithms have been proposed for
constructing cluster graphs along the spectrum (e.g. LTRIP Streicher and du Preez [2017]) (Fig. 4). Notably, join-graph
structuring Mateescu et al. [2010] spans the whole spectrum because it is controlled by a user-defined maximum cluster
size k∗, which can be varied from its smallest possible value to a value large enough to obtain a clique tree.
At the other end of the spectrum, the best maximum clique size
k
is
1 + tw(Gm)
, where
tw(Gm)
is the treewidth of
the moralized graph. Loopy BP becomes interesting when
tw(Gm)
is large, making exact BP costly. Unfortunately,
determining the treewidth of a general graph is NP-hard Arnborg et al. [1987], Bodlaender and Koster [2010]. Heuristics
such as greedy minimum-fill or nested dissection Strasser [2017], Hamann and Strasser [2018] can be used to obtain
clique trees whose maximum clique size kis near the optimum 1 + tw(Gm).
Different cluster graph algorithms could potentially be applied to the different biconnected components, or blobs
Gusfield et al. [2007] (e.g. LTRIP for one blob, clique tree for another), perhaps based a blob’s attributes that are easy
to compute. To choose between loopy versus exact BP, or between different cluster graph constructions more generally,
one could use traditional complexity measures of phylogenetic networks as potential predictors of cost-effectiveness.
For example, the reticulation number
h
is straightforward to compute. In a binary network, where all internal non-root
nodes have degree 3,
h
is simply the number of hybrid nodes. More generally
h=|{hybrid edges}|− |{hybrid nodes}|
Van Iersel et al. [2010]. The level of a network is the maximum reticulation number within a blob Gambette et al.
[2009]. The network’s level ought to predict treewidth better than
h
because a graph’s treewidth equals the maximum
treewidth of its blobs Bodlaender [1998], and moralizing the network does not affect its nodes’ blob membership. These
phylogenetic complexity measures do not predict treewidth perfectly Scornavacca and Weller [2022] except in simple
cases as shown below, proved in SM section B.
Proposition 1. Let
N
be a binary phylogenetic network with
h
hybrid nodes, level
ℓ
, and let
t
be the treewidth of the
moralized network
Nm
obtained from
N
. For simplicity, assume that
N
has no parallel edges and no degree-2 nodes
other than the root.
(A0) If ℓ= 0 then h= 0 and t= 1.
(A1) If ℓ= 1 then h≥1and t= 2.
(A2)
Let
v1
be a hybrid node with non-adjacent parents
u1, u2
. If
v1
has a descendant hybrid node
v2
such that one
of its parents is not a descendant of either u1or u2, then ℓ≥2and t≥3.
Level-1 networks have received much attention in phylogenetics because they are identifiable under various models
under some mild restrictions Sol
´
ıs-Lemus and An
´
e [2016], Ba
˜
nos [2019], Gross et al. [2021], Xu and An
´
e [2023].
Several inference methods limit the search to level-1 networks Sol
´
ıs-Lemus and An
´
e [2016], Oldman et al. [2016],
Allman et al. [2019], Kong et al. [2022]. Since moralized level-1 networks have treewidth 2, exact BP is guaranteed to
be efficient on them.
17

Leveraging graphical model techniques to study evolution on phylogenetic networks
3
10
30
100
300
1 10 100 1000
h: number of hybrids
Max cluster size upper bound
n: number of tips
2
10
25
35
50
100
1850
A
3
10
30
100
300
1 10 100 1000
level
Max cluster size upper bound
Network
empirical
simulated
B
Figure 6: We observe a positive sublinear relationship between a maximum clique size upperbound (from the greedy
minfill heuristic) and the number of hybrids (A) or network level (B) on a combined sample of 11 empirical networks
and 2509 simulated birth-death-hybridization networks. The empirical networks were sampled from [Maier et al., 2023,
Figs. 3a-c (left), 4a-c (left)] (reported as estimated by Bergstr
¨
om et al. [2020], Librado et al. [2021], Hajdinjak et al.
[2021], Lipson et al. [2020], Wang et al. [2021], Sikora et al. [2019]), [Lazaridis et al., 2014, Fig. 3], [Nielsen et al.,
2023, Fig. 3 (left)], [Sun et al., 2023, Fig. 4c], [M
¨
uller et al., 2022, Fig. 1a], [Neureiter et al., 2022, Fig. 5a]; fit by
these authors using
ADMIXTOOLS
Patterson et al. [2012], Maier et al. [2023],
admixturegraph
Lepp
¨
al
¨
a et al. [2017],
OrientAGraph
Molloy et al. [2021],
contacTrees
Neureiter et al. [2022],
Recombination
M
¨
uller et al. [2022],
AdmixtureBayes
Nielsen et al. [2023]. The simulated networks were obtained by subsampling 10 networks per
parameter scenario simulated by Justison and Heath [2024], then filtering out networks of treewidth 1 (trees, possibly
with parallel hybrid edges).
Beyond level-1, a network has a hybrid ladder (also called stack Semple and Simpson [2018]) if a hybrid node
v1
has a
hybrid child node
v2
. By Proposition 1, a hybrid ladder has the potential to increase treewidth of the moralized network
and decrease BP scalability, if the remaining conditions in (A2) are met. Related results in Chaplick et al. [2023] are
for undirected graphs that do not require prior moralization, and contain ladders defined as regular
2×L
grids. Their
Observation 1, that a graph containing a non-disconnecting grid ladder of length
L≥2
has treewidth at least 3, relies
on a similar argument as for (A2). However, structures leading to the conditions in (A2) are more general, even before
moralization. It may be interesting to extend some of the results from Chaplick et al. [2023] to moralized hybrid ladders
in rooted networks.
In Fig. 5 (right)
N2
has a hybrid ladder that does not meet all conditions of (A2), and has
t= 2
. Generally, outerplanar
networks have treewidth at most
2
Bodlaender [1998], and if bicombining (hybrid nodes have exactly 2 parents), remain
outerplanar after moralization. Networks in which no hybrid node is the descendant of another hybrid node in the same
blob are called galled networks Huson et al. [2010]. They provide more tractability to solve the cluster containment
problem Huson et al. [2009]. Here, galled networks would then never meet the assumptions of (A2) and it would be
interesting to study their treewidth after moralization.
We performed an empirical investigation of how
h
and
ℓ
can predict the treewidth
t
of the moralized network. Fig. 6
shows that
t
correlates with
h
and
ℓ
, on networks estimated from real data using various inference methods and on
networks simulated under the biologically realistic birth-death-hybridization model Justison and Heath [2024], Justison
et al. [2023], especially for complex networks. For networks with hundreds of tips (Thorson et al. [2023] lists several
studies of this size), large maximum clique sizes
k≥30
are not uncommon. In contrast, a Bethe cluster graph would
have maximum cluster size
k∗= 3
, so that
(k/k∗)3≥103
would provide a large computational gain for loopy BP to
be considered.
5.3.2 Approximation quality with loopy BP
We simulated data on a complex graph (40 tips, 361 hybrids) [M
¨
uller et al., 2022, Fig. 1a] and a simpler graph (12 tips,
12 hybrids) [Lipson et al., 2020, Extended Data Fig. 4], then compared estimates from exact and loopy BP. For both
networks, edges of length 0 were assigned the minimum non-zero edge length after suppressing any non-root degree-2
nodes. Trait values
x= (x1,...,xn)
at the tips were simulated from a BM with rate
σ2= 1
and
xρ= 0
at the root.
Figure 7 shows the exact and approximate log-likelihood and conditional mean and variance of
xρ
assuming a BM
with rate
σ2= 1
but improper prior
xρ∼ N(0,∞)
, using a greedy minimum-fill clique tree
U
and a cluster graph
U∗
. Using a factor graph, calibration failed for the complex network (SM section C, Fig. S2), so we used join-graph
structuring to build
U∗
.
U
can be calibrated in one iteration and the calculated quantities are exact (horizontal lines). In
contrast,
U∗
requires multiple passes and gives approximations. Calibration required more iterations on the complex
18

Leveraging graphical model techniques to study evolution on phylogenetic networks
network (
h= 361
) than on the simpler network (
h= 11
), as expected. But for both networks, the factored energy
(8)
approximated the log-likelihood very well. The distribution of the root state
xρ
conditional on the data seems
more difficult to approximate. The conditional mean was correctly estimated but required more iterations than the
log-likelihood approximation on the complex network. The conditional variance was severely overestimated on the
complex network and very slightly overestimated on the simpler network. As desired, the average computing time per
belief update was lower on
U∗
, although modestly so due to the clique tree
U
having many small clusters of size similar
to those in U∗(Fig. S3).
5 10 15 20
-2
-1
0
1
2
clique tree
join-graph str, R1
join-graph str, R2
5 10 15 20
0
5
10
15
20
25
30
5 10 15 20
-80
-70
-60
-50
-40
0 50 100 150 2 00
-60
-40
-20
0
20
0 10 20 30 40 5 0
0
500
1000
1500
2000
2500
0 10 20 30 40 5 0
-600
-500
-400
-300
-200
-100
Number of iterations
E(Xρ |data)
E(Xρ |data)
Var(Xρ|data)
Var(Xρ|data)
Factored energy
Factored energy
Lipson et al. (2020b): n=12, ℓ=12, h=12, k=6, k∗=3
Müller et al. (2022): n=40, ℓ=358, h=361, k=54, k∗=10
Figure 7: Accuracy of loopy BP. Approximation of the conditional distribution of the root state
Xρ
(left and center) and
log-likelihood (right) using a greedy minimum-fill clique tree Uand a join-graph structuring cluster graph U∗for two
networks of varying complexity M
¨
uller et al. [2022], Lipson et al. [2020] as measured by their number of tips (
n
), level
(
ℓ
), number of hybrids (
h
), maximum clique size (
k
), and maximum cluster size (
k∗
). For
U
, estimates are exact after
one iteration and shown as horizontal red lines. For
U∗
, estimates are shown over 20 (first row), 50 or 200 (second row)
iterations. Each iteration consists of two passes through each spanning tree in a minimal set that jointly covers
U∗
. In
each plot, the two curves correspond to two different regularizations of initial beliefs (SM section E, dotted: algorithm
R1, solid: algorithm R2).
6 Leveraging BP for efficient parameter inference
6.1 BP for fast likelihood computation
In some particularly simple models, such as the BM on a tree, fast algorithms such as IC Felsenstein [1985] or
phylolm
Ho and An
´
e [2014] can directly calculate the best-fitting parameters that maximize the restricted likelihood (REML), in
a single tree traversal avoiding numerical optimization. For more general models, such closed-form estimates are not
available. One product of BP is the likelihood of any fixed set of model parameters. BP can hence be simply used as a
fast algorithm for likelihood computation, which can then be exploited by any statistical estimation technique, in a
Bayesian or frequentist framework. Most of the tools cited in section 2.3 use either direct numerical optimization of the
likelihood Mitov et al. [2019], Boyko et al. [2023], Bartoszek et al. [2023] or sampling techniques such as Markov
Chain Monte Carlo (MCMC) Pybus et al. [2012], FitzJohn [2012] for parameter inference.
BP also outputs the trait distribution at internal, unobserved nodes conditioned on the observed data at the tips. In
addition to providing a tool for efficient ancestral state reconstruction, these conditional means and variances can be
used for parameter inference, with approaches based on latent variable models such as Expectation Maximization (EM)
Bastide et al. [2018a], or Gibbs sampling schemes Cybis et al. [2015]. Although not currently used in the field of
19

Leveraging graphical model techniques to study evolution on phylogenetic networks
evolutionary biology to our knowledge, approaches based on approximate EM algorithms Heskes et al. [2003] and
relying on loopy BP could also be used.
6.2 BP for fast gradient computation
As we show below, the conditional means and variances at ancestral nodes can be used to efficiently compute the
gradient of the likelihood Salakhutdinov et al. [2003]. The gradient of the likelihood can help speed up inference in many
different statistical frameworks Barber [2012]. In a phylogenetic context, gradients have been used to improve maximum
likelihood estimation Ji et al. [2020], Bayesian estimation through Hamiltonian Monte Carlo (HMC) approaches Zhang
et al. [2021], Fisher et al. [2021], Bastide et al. [2021], or variational Bayes approximations Fourment and Darling
[2019]. Although automatic differentiation can be used on trees for some models Swanepoel et al. [2022], direct
computations of the gradient using BP-like algorithms have been shown to be more efficient in some contexts Fourment
et al. [2023]. After recalling Fisher’s identity to calculate gradients after BP calibration, we illustrate its use on the
BM model (univariate or multivariate) where it allows for the derivation of a new analytical formula for the REML
parameter estimates.
6.2.1 Gradient Computation with Fisher’s Identity
In a phylogenetic context, latent variables are usually internal nodes, while observed variables are leaves. We write
Y={Xv,j :trait jobserved at v∈V}
the set of observed variables. Fisher’s identity provides a way to link
the gradient of the log-likelihood of the data
LL(θ) = log pθ(Y)
at parameter
θ
, with the distribution of all the
variables conditional on the observations Y. We refer to [Capp´
e et al., 2005, chap. 10] or [Barber, 2012, chap. 11] for
general introductions on Markov models with latent variables. Under broad assumptions, Fisher’s identity states (see
Proposition 10.1.6 in Capp´
e et al. [2005], or Section 11.6 in Barber [2012]):
∇θ′[log pθ′(Y)]|θ′=θ= Eθ[∇θ′[log pθ′(Xv;v∈V)]|θ′=θ|Y],
where
∇θ′[f(θ′)]|θ′=θ
denotes the gradient of
f
with respect to the generic parameters
θ′
and evaluated at
θ′=θ
, and
Eθ[• | Y]
the expectation conditional on the observed data under the model parametrized by
θ
, which is precisely
where the output from BP can be used. Plugging in the factor decomposition from the graphical model (4) we get:
∇θ′[log pθ′(Y)]|θ′=θ=X
v∈V
Eθ[∇θ′[log ϕv(Xv|Xu, θ′;u∈pa(v))]|θ′=θ|Y].(9)
While
(9)
applies to the full vector of all model parameters, it can also be applied to take the gradient with respect to a
single parameter
θ
of interest, keeping the other parameters fixed. For instance, we can focus on one rate matrix
Σ
of
a BM model, or one primary optimum of an OU model. Special care needs to be taken for gradients with respect to
structured matrices, such as variance matrices that need to be symmetric (see e.g. Bastide et al. [2021]) or with a sparse
inverse under structural equation modeling for high dimentional traits Thorson and van der Bijl [2023].
For models where the conditional expectation of the factor in
(9)
has a simple form, this formula is the key to an
efficient gradient computation. In particular, for discrete traits as in Example 2, the expectation becomes a sum of a
manageable number of terms, local to a cluster, weighted by the normalized cluster belief after calibration [Koller and
Friedman, 2009, ch. 19].
6.2.2 Gradient computation for linear Gaussian models
For linear Gaussian models
(2)
, log-factors can be written as quadratic forms
(6)
, so their derivatives have analytical
formulas (see SM section D). The conditional expectation in
(9)
then only depends on the joint first and second order
moments of the variables
(Xv, Xpa(v))
in a cluster, which are known as soon as the beliefs are calibrated. When the
graph is a tree, Bastide et al. [2021] exploited this formula to derive gradients in the general linear Gaussian case.
However, they did not use the complete factor decomposition
(4)
, but instead an ad-hoc decomposition that only works
when the graph is a (binary) tree, and exploits the split partitions defined by the tree. In contrast, the present approach
gives a recipe for the efficient gradient computation of any linear Gaussian model on any network, as soon as beliefs are
calibrated.
In the special case where the process is a homogeneous BM (univariate or multivariate) on a network with a weighted-
average merging rule
(3)
, a constant rate
Σ
, no missing data at the tips, and, if present, within-species variation that is
proportional to
Σ
, then the gradient with respect to
Σ
takes a particularly simple form. Setting this gradient to zero,
we find an analytical formula for the REML estimate of
Σ
and for the ML estimate of the ancestral mean
µρ
(SM
section D.3). In a phylogenetic regression setting, a similar formula can be found for the ML estimate of coefficients
(SM section D.4). Efficient algorithms such as IC and
phylolm
already exist to compute these quantities on a tree, in a
20

Leveraging graphical model techniques to study evolution on phylogenetic networks
single traversal. Here, our formulas need two traversals but remain linear in the number of tips, and because they rely
on a general BP formulation, they apply to networks with reticulations. Fisher’s identity and BP hence offer a general
method for gradient computation, and could lead to analytical formulas for other simple models. Such efficient formulas
could alleviate numerical instabilities observed in software such as
mvSLOUCH
, which experienced a significant failure
rate for the BM on trees with a large number of traits Bartoszek et al. [2024].
6.2.3 Hessian computation with Louis’s identity
Using similar techniques, the Hessian of the log-likelihood with respect to the parameters can also be obtained as a
conditional expectation of the Hessian of the complete log-likelihood:
n∇2
θ′[log pθ′(Y)] + ∇θ′[log pθ′(Y)] [∇θ′[log pθ′(Y)]]⊤oθ′=θ=
Eθh∇2
θ′[log pθ′(Xv;v∈V)] + ∇θ′[log pθ′(Xv;v∈V)] [∇θ′[log pθ′(Xv;v∈V)]]⊤|θ′=θYi.
This so-called Louis identity Capp
´
e et al. [2005] also simplifies under the factor decomposition
(4)
, and leads to
tractable formulas in simple Gaussian or discrete cases.
6.3 BP for direct Bayesian parameter inference
Likelihood or gradient-based approaches require careful analytical computations to get exact formulas in any new
model within the class of linear Gaussian graphical models, depending on the parameters of interest Bastide et al.
[2021]. One way to alleviate this problem is to use a Bayesian framework, and expand the graphical model to include
both the phylogenetic network and the evolutionary parameters, which are seen as random variables themselves, as
e.g., in H
¨
ohna et al. [2014]. Then, inferring parameters amounts to learning their conditional distribution in this larger
graphical model. In this setting, the output of interest from BP is not the likelihood but the distribution of random
variables (evolutionary parameters primarily) conditional on the observed data.
Exact computation may not be possible in this extended graphical model, because it is typically not linear Gaussian
and the graph’s treewidth can be much larger than that of the phylogenetic network, when one parameter (e.g. the
evolutionary rate) affects multiple node families. Therefore, approximations may need to be used. For example,
“black box” optimization techniques rely on variational approaches to reach a tractable approximation of the posterior
distribution of model parameters Ranganath et al. [2014]. The conditional distribution of unobserved variables, provided
by BP, facilitates the noisy approximation of the variational gradient that can be used to speed up the optimization of
the variational Bayes approximation.
7 Challenges and Extensions
7.1 Degeneracy
While our implementation provides a proof-of-concept, various technical challenges still need to be solved. Much of
the literature on BP focuses on factor graphs, which failed to converge for one of our example phylogenetic networks.
More work is needed to better understand the convergence and accuracy of alternative cluster graphs, and on other
choices that can substantially affect loopy BP’s efficiency, such as scheduling. Below, we focus on implementation
challenges due to degeneracies.
For the message
˜µi→j
to be well-defined in step 1 of Gaussian BP, the belief of the sending cluster must have a precision
matrix
K
in
(6)
with a full-rank submatrix with respect to the variables to be integrated out (
KI
in Algorithm 2). This
condition can fail under realistic phylogenetic models, due to two different types of degeneracy.
The first type arises from deterministic factors: when
Vv= 0
in
(2)
and
Xv
is determined by the states at parent nodes
Xpa(v)
without noise, e.g. when all of
v
’s parent branches have length 0 in standard phylogenetic models. This is
expected at hybridization events when both parents have sampled descendants in the phylogeny, because the parents and
hybrid need to be contemporary of one another. This situation is also common in admixture graphs Maier et al. [2023]
due to a lack of identifiability of hybrid edge lengths from
f
statistics, leading to a “zipped-up” estimated network
in which the estimable composite length parameter is assigned to the hybrid’s child edge Xu and An
´
e [2023]. With
this degeneracy,
Xv
has infinite precision given its parents, that is,
K
has some infinite values. The complications are
technical, but not numerical. For example, one can use a generalized canonical form that includes a Dirac distribution
to capture the deterministic equation of
Xv
given
Xpa(v)
from
(2)
. Then BP operations need to be extended to these
generalized canonical forms, as done in Schoeman et al. [2022] (illustrated in SM section F). One could also modify the
21

Leveraging graphical model techniques to study evolution on phylogenetic networks
network by contracting internal tree edges of length 0. At hybrid nodes, adding a small variance to
Vv
would be an
approximate yet biologically realistic approach.
The second type of degeneracy arises when the precision submatrix
KI
is finite but not of full rank. In phylogenetic
models, this is frequent at initialization
(5)
. For example, consider a cluster of 3 nodes: a hybrid
v
and its 2 parents. By
(7)
we see that
rank(Kv)≤p
. So at initialization with belief
ϕv
,
KI
is degenerate if we seek to integrate out
|I|= 2
nodes, which would occur if the cluster is adjacent to a sepset containining only one parent of
v
. This situation is typical
of factor graphs. Initial beliefs would also be degenerate with
K=0
for any cluster that is not assigned any factor by
(5)
. This may occur if there are more clusters than node families, or if the graph has nested redundant clusters (e.g.
from join-graph structuring). In some cases, a schedule may avoid these degeneracies, guaranteeing a well-defined
message at each BP update. On a clique tree, a schedule based on a postorder traversal has this guarantee, provided
that all
p
traits are observed at all leaves or that trait
j
at node
v
is removed from scope if it is unobserved at all its
descendants. But generally, it is unclear how to find such a schedule. Another approach is to simply skip a BP update if
its message is ill-defined, though there is no guarantee that the sending cluster will eventually have a well-behaved
belief to pass the message later. A robust option is to regularize cluster beliefs, right after initialization
(5)
or during BP,
by increasing some diagonal elements of
K
to make
KI
of full rank. To maintain the probability model, this cluster
belief regularization is balanced by a similar modification to a corresponding sepset. SM section E describes two such
approaches that appear to work well in practice, although theoretical guarantees have not been established.
7.2 Loopy BP is promising for discrete traits
We focused on Gaussian models in this paper, for which the ‘lazy’ matrix approach is polynomial. For discrete trait
models, the computational gains from loopy BP can be much greater, because alternative approaches are not polynomial
on general networks. For a trait with
c
states (
c= 2
for a binary trait as in Example 2), passing a message has complexity
O(ck)
where
k
is the sending cluster size. Thus, cluster graphs with small clusters can bring exponential computational
gains. Even exact BP can bring significant computational gains to existing approaches that rely on other means to
reduce complexity. For example, the model without ILS used in Lutteropp et al. [2022], Allen-Savietta [2020] is a
mixture model, so the network likelihood can be calculated as a weighted average of tree likelihoods for which exact
BP takes linear time. This approach scales exponentially with
h
because there are typically
O(2h)
trees displayed in a
network. In contrast, the complexity of BP on a clique tree of maximum clique size
k
is
O(nck)
, thus parametrized
by the treewidth
t
of the moralized network instead of
h
(
t=k−1
for an optimal clique tree). Given our empirical
evidence that
t
grows more slowly than
h
or the network’s level
ℓ
in biologically-realistic networks (Fig. 6), exact BP
could achieve significant computational gains and loopy BP substantially more.
A BP approach is already used in
momi2
Kamm et al. [2020], who use a clique tree built from a node ordering by
age from youngest to oldest, to get conditional likelihoods of the derived allele count under a Moran model (without
mutation). The mutation-with-ILS model in
SnappNet
can be also reframed as a graphical model on a graph expanded
from the phylogenetic network (as shown in Example 3 and SM section A). Accordingly, the BP-like algorithm in
Rabier et al. [2021] has complexity controlled by the network’s scanwidth, a parameter introduced by Berry et al.
[2020]. Using regular BP on more optimal clique trees and loopy BP on cluster graphs may help speed up computations
even more.
Also related is the algorithm in Scornavacca and Weller [2022], who use a clique tree to solve a parsimony problem. In
this non-probabilistic setting, it is unclear how cluster graphs could be leveraged to speed up algorithms as they do in
loopy BP.
To deal with computational intractability, the most widely-used probabilistic methods to infer networks from DNA
sequences are based on composite likelihoods Sol
´
ıs-Lemus and An
´
e [2016], Yu and Nakhleh [2015] or summary
statistics like
f
statistics Maier et al. [2023], Nielsen et al. [2023], leading to a lack of identifiability for parts of the
network topology and some of its parameters Sol
´
ıs-Lemus and An
´
e [2016], Ba
˜
nos [2019], Xu and An
´
e [2023], An
´
e
et al. [2024], Allman et al. [2024], Rhodes et al. [2024]. These identifiability issues should be alleviated if using the full
data becomes tractable thanks to exact or loopy BP.
Supplementary Material
Technical derivations are available in the Supplementary Material (SM). Code to reproduce Figures 4a, 6 and 7 is avail-
able at
https://github.com/bstkj/graphicalmodels_for_phylogenetics_code
. A julia package for Gaus-
sian BP on phylogenetic networks is available at
https://github.com/cecileane/PhyloGaussianBeliefProp.
jl.
22

Leveraging graphical model techniques to study evolution on phylogenetic networks
Acknowledgements
This work was supported in part by the National Science Foundation (DMS 2023239 to C.A.) and by the University
of Wisconsin-Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the
Wisconsin Alumni Research Foundation. C.A. visited P.B. at the University of Montpellier thanks to support from the
I-SITE MUSE through the Key Initiative “Data and Life Sciences”.
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30

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
SUP PLEMEN TARY MATERI AL
A Recasting SnappNet as BP
SnappNet
[Rabier et al., 2021] extends the model described in
SNAPP
[Bryant et al., 2012] to binary phylogenetic
networks with reticulations. In the main text, their model is considered along a 2-taxon phylogenetic tree and Fig. 3b
shows that the graphical model has a more complicated graph. The same applies in the presence of reticulations (see
Fig. S1 for a 2-taxon phylogeny with 1 reticulation). In addition to the coalescent and speciation factors described
in Example 3, we also need to describe hybridization factors. Consider an edge
e
that is the child of a hybrid node,
whose parent hybrid edges
p1
and
p2
have inheritance probabilities
γ1
and
γ2
. The hybridization factors for the total
allele count
ϕnp1=P(np1|ne)
and
ϕnp2=
1
{ne−np1}(np2)
describe a binomial distribution for each
npi
(
i= 1,2
)
with
np1+np1=ne
, because each of the
ne
individuals has a
γi
chance of being assigned to edge
pi
(
i= 1,2
). The
hybridization factor for the red allele count is simply ϕre=
1
{rp1+rp2}(re)because re=rp1+rp2.
(a)
ρ, ρ
1,12,2
5
(b)
5ρρ44
55 3 344
113322
11 2 2
(c)
5,3,5,3
5,3,5,3
C3
1,5,3
1,5,3
C2
1,1
1,1
C1
5,4, ρ
5,4,ρ
C7
5,4,4
5,4,4
C6
5,3,2,4
5,3,2,4
C5
2,2
2,2
C4
1,1
5,5
γ= 0.6
5,53,3
3,3γ= 0.42,2
4,4
4,4
1
1
S1,2
5,3
5,3
S2,3
5,3
5,3
S3,5
2
2
S4,5
5,4
5,4
S5,6
5,4
5,4
S6,7
Figure S1: (a) Phylogenetic network
N
with hybrid edges in blue. (b) Graph
G
for the graphical model associated with
N
.
G
is a DAG with two roots (
1
and
2
) and two leaves (
1
and
2
). (c) Clique tree
U
for
G
, with clusters
Cj
in grey and
sepsets
Si,j
in orange. To reduce clutter and simplify notations in this figure,
ne
and
re
are both abbreviated as
e
and
are distinguished by colours (
n
’s in black,
r
’s in red). Similarly,
ne
and
re
are both denoted as
e
and distinguished by
colours.
We show that applying the
SnappNet
algorithm to the network in Fig. S1(a) is equivalent to BP on the clique tree
in Fig. S1(c). To start, we assign the initial beliefs
ϕn1ϕr1
to cluster
C1
,
ϕn2ϕr2
to
C4
,
ϕnρϕrρϕr5ϕr4
to
C7
;
ϕn5ϕn3ϕr5ϕr3
to
C3
,
ϕn4ϕr3ϕr2
to
C5
, and
ϕn4ϕr4
to
C6
. Finally, all hybridization factors
ϕn5ϕn3ϕr1
are assigned to
C2
. Each sepset is assigned an initial belief of 1. The total allele counts
n1
,
n2
(fixed by design) and the observed red
allele counts
r1
,
r2
at the tips are absorbed as evidence into the mutation factors
ϕr1
,
ϕr2
and coalescent factors
ϕn1
,
ϕn2
for the terminal edges. This is denoted as
ϕ[·]
, with
[·]
containing the evidence absorbed. BP messages are then
31

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
passed on the clique tree from C1and C4(considered as leaves) towards C7(considered as root) as follows:
˜µ1→2=ϕn1[n1]ϕr1[n1,r1] = F1(S1,2)
˜µ2→3=X
C2\S2,3
ϕn5ϕn3ϕr1·˜µ1→2= F5,3(S2,3)
˜µ3→5=X
C3\S3,5
ϕn5ϕn3ϕr5ϕr3·˜µ2→3= F5,3(S3,5)
˜µ4→5=ϕn2[n2]ϕr2[n2,r2] = F2(S4,5)
˜µ5→6=X
C5\S5,6
ϕn4ϕr3ϕr2·˜µ3→5˜µ4→5= F5,4(S5,6)
˜µ6→7=X
C6\S6,7
ϕn4ϕr4·˜µ5→6= F5,4(S6,7)
βfinal
7=ϕnρϕrρϕr5ϕr4·˜µ6→7
X
C7
βfinal
7=X
C7\S6,7
ϕrρX
S6,7
ϕnρϕr5ϕr4·˜µ6→7=X
C7\S6,7
ϕrρFρ(C7\ S6,7)
where the the
Fz
functions are defined in Rabier et al. [2021] for different population interface sets
z
(each population
interface is the top or bottom of some branch, e.g.
ρ
,
ρ
). That this correspondence does not always hold (e.g. suppose
the messages were sent towards
C1
instead), highlights that BP is more general. The
Fz
s recursively compose the
likelihood according to rules described in Rabier et al. [2021], and can be expressed at the top-level as
Pnρ,rρϕrρFρ
.
This is precisely the quantity from marginalizing βfinal
7above, the final belief of C7.
B Bounding the moralized network’s treewidth
Proof of Proposition 1.
Using the notations in the main text, let
N
be as a binary phylogenetic network with
h
hybrid
nodes, level
ℓ
, no parallel edges and no degree-2 nodes other than the root. Let
t
be the treewidth of the moralized graph
Nmobtained from N.
(A0) is well-known:
t= 1
exactly when
N
is a tree. If
ℓ= 1
then
N
has at least one non-trivial blob and every such
blob is a cycle. So Nmhas outerplanar blobs and t= 2 Biedl [2015], proving (A1).
Now consider hybrid nodes
v1
and
v2
as in (A2). Let
u3, u4
be the parents of
v2
such that
u3
is not a descendant of
u1
or
u2
(see Fig. 5, in which
u4=v1
). Then there must be a path
p4
from
v1
to
v2
through
u4
since
v2
is a descendant of
v1
. For a directed path
p
, let
pu
denote the corresponding undirected path. Let
w
be a strict common ancestor of
u1
and
of
u2
such that there exist disjoint paths
p1
,
p2
, with
pi
from
w
to
ui
. Such
w
exists because
u1
has a parent other than
u2
and vice versa. Let
C
be the cycle in
Nm
formed by concatenating
pu
1
,
pu
2
and the moral edge
{u1, u2}
. Next, pick
any path
p
from the root of
N
to
u3
. If
p
does not share any node with
C
, then we can find a common ancestor
˜w
of
w
and
u3
, and paths
pw
and
p3
from
˜w
to
w
and
u3
respectively, that do not intersect
p1
nor
p2
. Then we can see that
Nm
contains the complete graph on
{w, u1, u2, v1}
as a graph minor, by contracting
pu
w+pu
3+{u3, v2}+pu
4
into
a single edge between
w
and
v1
. If instead
p
intersects
C
, then let
w′
be the lowest node at which
p
and
C
intersect.
Then
w′=u1
because otherwise
u3
would be its descendant. Similarly
w′=u2
. Let
p3
denote the subpath of
p
from
w′to u3. Then Nmcontains the complete graph on {w′, u1, u2, v1}as a graph minor, as Ccan be contracted into the
cycle
{w′, u1, u2}
and
pu
3+{u3, v2}+pu
4
can be contracted into an edge
{w′, v1}
. In both cases,
Nm
contains the
complete graph on 4 nodes as a graph minor, therefore its treewidth is
t≥3
Bodlaender [1998]. Also, in both cases
v1
and v2are in a common undirected cycle in N, so in the same blob and ℓ≥2.
32

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
C Approximation quality with loopy BP
5 10 15 20
-2
-1
0
1
2
clique tree
join -grap h str
factor graph
5 10 15 20
0
5
10
15
20
25
30
5 10 15 20
-80
-70
-60
-50
-40
0 50 100 150 200
-80
-60
-40
-20
0
20
0 10 20 30 40 50
0
500
1000
1500
2000
2500
0 10 20 30 40 50
-600
-500
-400
-300
-200
-100
0 50 100 150 200
-1012
-108
-104
0
104
108
1012
0 10 20 30 40 50
-102
-10
0
10
102
104
0 10 20 30 40 50
-104
-102
-10
-2
0
2
10
Number of iterations
E(Xρ |data)
E(Xρ |data)
Δ=E(Xρ |data)−15.7
Var(Xρ |data)
Var(Xρ |data)
Δ=Var(Xρ |data)−1425.7
Factored energy
Factored energy
Δ=Factored energy −(−85)
Lipson et al. (2020b): n=12
Müller et al. (2022): n=40
Müller et al. (2022): n=40, difference from clique tree estimate shown on log-modulus transformation scale
Figure S2: Comparing the accuracy of loopy BP between different cluster graphs: built from join-graph structuring
U∗
as in Fig. 7 (black), or a factor graph (purple). For both, initial beliefs are regularized using algorithm R4. The true
values, obtained using a clique tree, are shown in red. The plots in the first row are for the simpler phylogenetic network,
and the plots in the other rows are for the complex phylogeny. The last row shows the difference
∆
between the loopy
BP estimate and the true value, displayed on the log-modulus scale using the transformation
sign(∆) log(1 + |∆|)
. For
the simpler phylogenetic network, convergence speed and accuracy are similar between
U∗
and the factor graph, which
is unsurprising given their similarly small cluster sizes (
≤3
). For the complex network, the factor graph did not reach
calibration as its iterates diverged for the conditional mean and factored energy.
33

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
4.9
5.8
8.3
11
1 2 5 10 20 50
Cluster size (log−scale)
Mean time (µs) per message
Müller (n=40)
Lipson (n=12)
U
U*
U
U*
Figure S3: Boxplots (with means as points) showing the dis-
tribution of cluster sizes in the join-graph structuring cluster
graph
U∗
and in the clique tree
U
from Fig. 7. The factor
graph has clusters of size between 1 and 3 (not displayed).
The time for 100 iterations (defined in Fig. 7) was bench-
marked over 20 replicates on a MacBook Pro M2 2022, and
divided by the number of messages per 100 iterations to ob-
tain an estimate of the mean time per belief update (vertical
axis).
D Gradient and parameter estimates under the BM
D.1 The homogeneous BM model
We consider here the simple case of a multivariate BM of dimension
p
on a network: with
Vv=ℓ(e)Σ
at a tree node
v
with parent edge
e
. At a hybrid node, we assume a weighted average merging rule as in (3.2) with a possible extra
hybrid variance proportional to
Σ
:
e
Vh=e
ℓ(h)Σ
for some scalar
e
ℓ(h)≥0
. To simplify equations, we define
e
ℓ(v)=0
if
vis a tree node. Then for each node v∈Vthe Gaussian linear model (3.1) simplifies to:
XvXpa(v)∼ N
X
u∈pa(v)
γuvXu;ℓ(v)Σ
,(SM-1)
with γuv the inheritance probability associated with the branch going from uto v, and
ℓ(v) = e
ℓ(v) + X
u∈pa(v)
γ2
uvℓ(eu→v).
At the root, we assume a prior variance proportional to
Σ
:
Xρ∼ N(µρ;ℓ(ρ)Σ)
which may be improper (and
degenerate) with infinite variance ℓ(ρ) = ∞or ℓ(ρ)=0.
This model can also accommodate within-species variation, by considering each individual as one leaf in the phylogeny,
whose parent node corresponds to the species to which the individual belongs. The edge
e
from the species to the
individual is assigned length
ℓ(e) = w
and variance proportional to
Σ
conditional on the parent node (species average):
ℓ(e)Σ
. This model, then, assumes equal phenotypic (within species) correlation and evolutionary (between species)
correlation between the ptraits. The derivations below assume a fixed variance ratio w, to be estimated separately.
All results in this section use this homogeneous BM model, and make the following assumption.
Assumption 1.At each leaf, the trait vector (of length
p
) is either fully observed or fully missing, i.e. there are no
partially observed nodes.
D.2 Belief Propagation
Gaussian BP Algorithm 2 can be applied in the simple BM case to get the calibrated beliefs, with two traversals of
a clique tree (or convergence with infinitely many traversals of a cluster graph). The following result states that the
conditional moments of all the nodes obtained from this calibration have a very special form. We will use it to derive
analytical formulas for the maximum likelihood estimators of the parameters of the BM.
Proposition 2. Assume the homogeneous BM
(SM-1)
and Assumption 1. The expectation of the trait at each node
conditional on the observed data does not depend on the assumed Σparameter. In addition, the conditional variance
matrix and the conditional covariance matrix of a node trait and any of its parent’s is proportional to Σ.
34

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
To prove this proposition, we need the following technical lemma, which we prove later.
Lemma 3. Consider a homogeneous
p
-dimensional BM on a network and Assumption 1. At each iteration of the
calibration, each cluster and sepset of snodes has a belief whose canonical parameters are of the form:
K=J⊗Σ−1and h= (Is⊗Σ−1)


m1
.
.
.
ms


= (Is⊗Σ−1)vec(M) = vec(Σ−1M)(SM-2)
for some
s×s
symmetric matrix
J
and vectors
mi
(
i= 1 . . . s
) of size
p
, where
M
is the
p×s
matrix with
mi
on
column
i
, and where
vec
denotes the vectorization operation formed by stacking columns. Further,
M
depends linearly
on the data Y(from stacking the trait vectors at the tips) and µρ, separately across traits, in the sense that
mi= (wi⊗Ip)µρ
Y.(SM-3)
for some
1×(n+ 1)
vector of weights
wi
.
J
and vectors
wi
(
i= 1, . . . , s
) are independent of the variance rate
Σ
, the
data Yand µρ. They only depend on the network, the chosen cluster graph, the chosen cluster or sepset in this graph,
and the iteration number.
Proof of Proposition 2.
First note that, for any belief with form
(SM-2)
, the mean of the associated normalized Gaussian
distribution can be expressed as follows. Let the vector µjof size pbe the mean for the node indexed j. Then



µ1
.
.
.
µs


=K−1h= (J−1⊗Ip)


m1
.
.
.
ms


= (J−1⊗Ip)vec(M) = vec(MJ−1).
Let Ebe the p×smatrix of means with µjon column j. Then the expression above simplifies to
E=M J−1.
Assume Lemma 3, from which we re-use notations here. Let
C
be a cluster, and
K
,
J
and
M
be its matrices from
(SM-2)
. For any node
v
in
C
, let
kC(v)
be the index of
v
in
C
’s matrices. Then, writing
[J−1]•k
for the
kth
column
vector of J−1, we get:
E [ Xv|Y] = µkC(v)=M[J−1]•kC(v)=Ev,(SM-4)
where
Ev
denotes the column of
E
for node
v
(i.e. the conditional expectation of its trait), and does not depend on
Σ
. Assuming that calibration is reached,
Ev
does not depend on the cluster
C
(or sepset) containing
v
. Further, note
that
(SM-4)
is exact on any cluster graph at calibration, not simply approximate, because we are using a Gaussian
graphical model [Weiss and Freeman, 1999].
Similarly, for nodes u, v in C:
var [Xv|Y]=[K−1]kC(v)kC(v)= [J−1]kC(v)kC(v)Σ,
cov [Xv, Xu|Y]=[K−1]kC(v)kC(u)= [J−1]kC(v)kC(u)Σ.(SM-5)
Therefore, their conditional variances and covariances are proportional to Σ.
In the following, with a slight abuse of notation, for any two nodes
u, v
in
C
, we will write
[J−1]uv = [J−1]kC(u)kC(v)
and [K−1]uv = [K−1]kC(u)kC(v)for the submatrices corresponding to the indices for uand vin C.
Since SM-4 requires inverting
J
, whose size
s
depends on the cluster, calculating the conditional means
Ev
has
complexity
O(s3)
typically. As the
J
and
M
matrices appearing in
(SM-4)
and
(SM-5)
do not depend on
Σ
by
Lemma 3, they can be computed by running BP with any Σvalue, and we have the following.
Corollary 4. The
J
and
M
matrices in Lemma 3, used in
(SM-4)
and
(SM-5)
, are obtained as a direct output of BP
using Σ=Ipto calibrate the cluster graph.
Using
(SM-3)
in Lemma 3 and the derivation of vectors
wi
at each BP update, given in the proof below, we obtain the
following result.
Corollary 5. For each cluster, the weights
wi
appearing in
(SM-3)
can be obtained alongside BP for any trait using
updates
(SM-7)
and
(SM-8)
below, until convergence of all
wi
weight vectors and
J
matrices. These quantities can
then be used to obtain conditional expectations and conditional (co)variances for any trait using
(SM-3)
,
(SM-4)
and (SM-5).
35

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
For example, this result implies that obtaining calibrated conditional expectations for a large-dimensional trait can be
done without handling large
p×p
matrices: by first calculating the
wi
’s and
J
for each cluster until convergence, and
then re-using them repeatedly for each of the ptraits separately (without re-calibration).
Proof of Lemma 3.
We now show that the properties stated in Lemma 3 hold for each factor at initialization, and
continue to hold after each step of Algorithm 2: belief initialization, evidence absorption, and propagation.
Factor Initialization. Using the notations from the main text, each factor
ϕv(xv|xpa(v))
has canonical form over its
full scope:
ϕv(xv|xpa(v)) = C xv
xpa(v);Kv, hv, gv.
For any internal node or any leaf vbefore evidence absorption, we get from (4.4) and (SM-1):
Kv=J⊗Σ−1and hv=0,where J=1
ℓ(v)1−γ⊤
−γ γγ⊤(SM-6)
and
γ⊤= (γuv;u∈pa(v)).
Hence all the node family factors have form
(SM-2)
at initialization and neither
J
nor any
mi=0depend on the data. We can initialize wi=0for each iin the factor’s scope.
At the root
ρ
, the formulas above still hold using that
pa(ρ)
is empty and
γ
has length
0
, for any
0< ℓ(ρ)<+∞
,
and also for
ℓ(ρ)=+∞
in which case
J= [0]
, independent of the data. For
hρ
, we have
hρ=1
ℓ(ρ)Σ−1µρ
, which
satisfies
(SM-2)
with
mρ=µρ/ℓ(ρ)
.
mρ
is linear in
µρ
and satisfies
(SM-3)
with
wρ= ( 1
ℓ(ρ),0,...,0)
(or simply
0
if
ℓ(ρ)=+∞
). If
ℓ(ρ)=0
, the root factor is not assigned to any cluster at initialization because it is instead handled
during evidence absorption below. This is because
ℓ(ρ)=0
implies that
Xρ
is fixed to the
µρ
value, and this is handled
similarly to leaves fixed at their observed values.
Belief Initialization. Now consider a cluster
C
that is assigned factors
ϕv
for
k≥0
nodes
{v1, . . . , vk}
, by (4.2).
Before this assignment, the belief for
C
(and for all sepsets) is set to the constant function equal to 1, which trivially
satisfies (SM-2) with J=0and every mi=0, and satisfies (SM-3) with every wi=0. To assign factor ϕvjto C, we
first extend scope of
ϕvj
to the scope of
C
(re-ordering the rows and columns of
Kvj
and
hvj
to match that of
C
). We
then multiply
C
’s belief by the extended factor. So we now prove that
(SM-2)
is preserved by these two operations:
extension and multiplication.
Belief Extension. Consider extending the scope of a belief with parameters
(K, h, g)
satisfying
(SM-2)
to include all
p
traits of an extra
(s+ 1)th
node. Without loss of generality, we assign these extra variables the last
p
indices. Then the
canonical parameters of the extended belief can be written as
˜
K=J 0
0 0⊗Σ−1and ˜
h= (Is+1 ⊗Σ−1)



m1
.
.
.
ms
0




and continue to be of form
(SM-2)
.
˜
K
continues to be independent of
Σ
and of the data. All
mi
vectors involved in
˜
h
continue to be independent of
Σ
and linear in the data according to
(SM-3)
: with
wi
unchanged for
i≤s
and
wi=0
for i=s+ 1.
Beliefs Product and Quotient. Next, if
(K, h, g)
and
(K′, h′, g′)
are the parameters of two beliefs on the same scope
satisfying Lemma 3, then their product also satisfies Lemma 3 because the canonical form of the product has parameters
(K+K′, h +h′, g +g′)
, and can be expressed with
(SM-2)
using
J+J′
and
mj+m′
j
.
(SM-3)
continues to hold
using weight vectors
wj+w′
j
. Similarly, the ratio of the two beliefs has parameters
(K−K′, h −h′, g −g′)
and
continues to satisfy Lemma 3.
Evidence Absorption. Assume that a belief satisfies Lemma 3, and that we want to absorb the evidence from one node
u
,
1≤u≤s
. This node
u
can be a leaf, or the root if
ℓ(ρ)=0
. If
u=ρ
then the data to be absorbed is
xu=µρ
. We
need to express the canonical form of the belief as a function of
x−u= [x⊤
1, . . . , x⊤
u−1, x⊤
u+1, . . . , x⊤
s]⊤
only, letting
the data
xu
appear in the canonical parameters. By Assumption 1,
xu
is of full length
p
, which maintains the block
structure. We have:
x⊤
1···x⊤
s(J⊗Σ−1)


x1
.
.
.
xs


=x⊤
−u(J−u⊗Σ−1)x−u+ 2 X
t=u
x⊤
uJutΣ−1xt+ x⊤
uJuuΣ−1xu
=x⊤
−u(J−u⊗Σ−1)x−u+ 2(J−u,u ⊗xu)⊤(Is−1⊗Σ−1)x−u+ x⊤
uJuuΣ−1xu,
36

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
where
J−u
is the
(s−1) ×(s−1)
matrix
J
without the row and column for
u
; and
J−u,u
is the
(s−1) ×1
column
vector of Jfor uwithout the row entry for u. Likewise:
m⊤
1···m⊤
s(Is⊗Σ−1)


x1
.
.
.
xs


=m⊤
−u(Is−1⊗Σ−1)x−u+m⊤
uΣ−1xu,
where m−uis similarly defined as x−u, so that the canonical form of the factor satisfies:
log C(x−u;K−u, h−u, g−u) = −1
2x⊤
−u(J−u⊗Σ−1)x−u+ (m−u−J−u,u ⊗xu)⊤(Is−1⊗Σ−1)x−u+g−u,
where g−udoes not depend on x−u,
K−u= (J−u⊗Σ−1)and h−u= (Is−1⊗Σ−1)(m−u−J−u,u ⊗xu),
have form
(SM-2)
.
J−u
continues to be independent of
Σ
and of the data and
µρ
. All
mi
vectors involved in
h−u
continue to be independent of
Σ
and linear in the data —with linear dependence on
xu
introduced in this step. Namely,
(SM-3) holds with weight vector wjupdated to:
wj−Ji,ueuwhere eu= (0,...,0,1,0, . . .)(SM-7)
is the basis (row) vector of Rn+1 with coordinate 1 at the position indexing tip u.
Note that, for a tip
v
with data on all
p
traits, we recover (4.4) for the factor associated with the external edge to
v
,
whose scope is reduced to xpa(v)after absorbing the evidence from xv(with s= 2):
Kv=1
ℓ(v)Σ−1, hv=1
ℓ(v)Σ−1xv,and gv=−1
2log |2πℓ(v)Σ|+1
ℓ(v)∥xv∥2
Σ−1.
Propagation. Next, we show that beliefs continue to satisfy Lemma 3 after any propagation step of Algorithm 2. The
first propagation step consists of marginalizing a belief, to calculate the message
˜µi→j
from cluster
i
to cluster
j
.
Suppose that a belief with parameters
(K, h, g)
satisfies
(SM-2)
, and that we marginalize out all traits of one or more
nodes in its scope. Let
I
be the indices corresponding to nodes (or their traits, depending on the context, with some
abuse of notation) to be marginalized and
S
the indices corresponding to the remaining nodes (or their traits). Then, the
marginal belief has canonical parameters (˜
K,˜
h)with:
˜
K=KS−KS,IK−1
IKI,S
=JS⊗Σ−1−JS,I⊗Σ−1JI−1⊗ΣJI,S⊗Σ−1
=JS−JS,IJI−1JI,S⊗Σ−1=˜
J⊗Σ−1
and
˜
h=hS−KS,IK−1
IhI=hS−JS,I⊗Σ−1JI−1⊗ΣhI=hS−JS,IJI−1⊗IphI=


Σ−1˜m1
.
.
.
Σ−1˜ms



where, for j∈S:
˜mj=mj−X
i∈IJS,IJI−1ji mi.
So (SM-3) holds with updated weights:
˜wj=wj−X
i∈IJS,IJI−1ji wi,(SM-8)
and the marginalized belief (message) is still of the form
(SM-2)
and continues to satisfy Lemma 3. The remaining
propagation steps consist of dividing the message by the current sepset belief; extending the resulting quotient to the
scope of the receiving cluster; and multiplying the receiving cluster’s current belief with the extended quotient. Each of
these steps was already proved to preserve the properties of Lemma 3, therefore the receiving cluster’s new belief still
satisfies Lemma 3. The sepset belief does too because it is updated with the message that was passed.
37

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
D.3 Gradient computation and analytical formula for parameter estimates
D.3.1 Gradients of factors
When the factors are linear Gaussian as in (3.1), their derivarive with respect to any vector of parameters
θ
can be
written as:
∇θlog ϕv(Xv|Xpa(v), θ)=∂
∂θ [qvXpa(v)+ωv]⊤V−1
vXv−qvXpa(v)−ωv
+1
2
∂vech(V−1
v)⊤
∂θ vech Vv−(Xv−qvXpa(v)−ωv)(Xv−qvXpa(v)−ωv)⊤,(SM-9)
where
vech
is the symmetric vectorization operation [Magnus and Neudecker, 1986]. In the BM case, (3.1) simplifies
to (SM-1), so that, for non-root nodes:
∇θlog ϕv(Xv|Xpa(v), θ)=∂
∂θ 
X
u∈pa(v)
γuvXp

⊤
ℓ(v)−1Σ−1
Xv−X
u∈pa(v)
γuvXp

+1
2
∂vech(ℓ(v)−1Σ−1)⊤
∂θ vech 

ℓ(v)Σ−
Xv−X
u∈pa(v)
γuvXp

Xv−X
u∈pa(v)
γuvXp

⊤

,
and, for the root ρ, assuming 0< ℓ(ρ)<+∞,
∇θ[log ϕρ(Xρ|θ)] = ∂[µρ]⊤
∂θ ℓ(ρ)−1Σ−1(Xρ−µρ)
+1
2
∂vech(ℓ(ρ)−1Σ−1)⊤
∂θ vech ℓ(ρ)Σ−(Xρ−µρ) (Xρ−µρ)⊤.
D.3.2 Estimation of µρ
Note that
µρ
has no impact on the model and needs not be estimated if
ℓ(ρ) = ∞
(improper flat prior). We assume here
that
0< ℓ(ρ)<+∞
, and will consider the case
ℓ(ρ) = 0
later. Only the root factor depends on
µρ
. Taking its gradient
with respect to µρ, we get:
∇µρ[log ϕρ(Xρ|θ)] = ℓ(ρ)−1Σ−1(Xρ−µρ).
To apply Fisher’s formula (6.1), we take the expectation Eθ[• | Y]of this gradient conditional on all the data Y:
∇µ′
ρ[log pθ′(Y)]µ′
ρ=µρ
= Eθ∇µ′
ρ[log ϕρ(Xρ|θ′)]µ′
ρ=µρY=ℓ(ρ)−1Σ−1(Eθ[Xρ|Y]−µρ).
Setting this gradient to 0, we get:
ˆµρ= Eθ[Xρ|Y] = Eρ(SM-10)
where
C
is any cluster containing
ρ
in its scope, and
J
and
M
are the matrices in Lemma 3 for its belief. Note that
by Lemma 3, this estimate is independent of the assumed
Σ
used during calibration. This procedure corresponds to
maximum likelihood estimation under the assumption that
ℓ(ρ)
is known. Under this model,
µρ
represents the ancestral
state at time
ℓ(ρ)
prior to the root node
ρ
, which is typically taken as the most recent common ancestor of the sampled
leaves. This is equivalent to considering an extra root edge of length
ℓ(ρ)
above
ρ
, whose parent node has ancestral
state
µρ
. Then
ˆµρ
is a maximum likelihood estimate of the ancestral state at
ρ
, or an approximation thereof if a cluster
graph is used instead of a clique tree. Note that, in a Bayesian setting, when fixing
ℓ(ρ)
to a given value, and fixing
µρ= 0
, this model can be seen as setting a Gaussian prior on the value at the root of the tree. This is the model used
e.g. in BEAST [Fisher et al., 2021].
D.3.3 Estimation of Σ
We now take the gradient with respect to the vectorized precision parameter
P= vech(Σ−1)
, of length
p(p+ 1)/2
.
For v=ρ, we get:
∇Plog ϕv(Xpa(v)|Xu, θ)=1
2ℓ(v)−1vech 

ℓ(v)Σ−
Xv−X
u∈pa(v)
γuvXu

Xv−X
u∈pa(v)
γuvXu

⊤


38

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
and, for the root ρ:
∇P[log ϕρ(Xρ|θ)] = 1
2ℓ(ρ)−1vech ℓ(ρ)Σ−(Xρ−µρ) (Xρ−µρ)⊤.
Applying again Fisher’s formula (6.1), we get:
∇P′[log pθ′(Y)]|P′=P=1
2X
v∈V
vech Σ−ℓ(v)−1Fv,
where
Fv
is derived next, using that
EZZ ⊤= var [Z] + E [Z] E [Z]⊤
and using
(SM-4)
and
(SM-5)
on a cluster
C
containing vand its parents in its scope, with Jvand Mvfrom Lemma 3 for C. For v=ρ, we get:
Fv= Eθ


Xv−X
u∈pa(v)
γuvXu

Xv−X
u∈pa(v)
γuvXu

⊤
Y


= varθ
Xv−X
u∈pa(v)
γuvXuY
+ Eθ
Xv−X
u∈pa(v)
γuvXuY
Eθ
Xv−X
u∈pa(v)
γuvXuY

⊤
=
[J−1
v]vv +X
u1,u2∈pa(v)
γu1vγu2v[J−1
v]u1u2−2X
u∈pa(v)
γuv[J−1
v]vu
Σ
+
Ev−X
u∈pa(v)
γuvEu

Ev−X
u∈pa(v)
γuvEu

⊤
.
For the root ρ:
Fρ= Eθh(Xρ−µρ) (Xρ−µρ)⊤Yi
= varθ[Xρ−µρ|Y]+Eθ[Xρ−µρ|Y] Eθ[Xρ−µρ|Y]⊤
= [J−1
ρ]ρρΣ+ (Eρ−µρ) (Eρ−µρ)⊤.
Setting this gradient to 0 with respect to Σ, we get the following maximum likelihood estimate for the rate matrix:
ˆ
Σ=
ℓ(ρ)−1(Eρ−ˆµρ) (Eρ−ˆµρ)⊤+X
v∈V,v=ρ
ℓ(v)−1
Ev−X
u∈pa(v)
γuvEu

Ev−X
u∈pa(v)
γuvEu

⊤

×
X
v∈V
1−ℓ(v)−1
[J−1
v]vv +X
u1,u2∈pa(v)
γu1vγu2v[J−1
v]u1u2−2X
u∈pa(v)
γuv[J−1
v]vu


−1
(SM-11)
where we use the convention that a sum over an empty set (here pa(ρ)) is 0.
Note that this formula only uses the calibrated moments computed at each cluster. After calibration, then, calculating
ˆ
Σ
with SM-11 has complexity
O(|V|(k3+p2))
where
k
is the maximum cluster size, since SM-11 requires inverting at
most
|V|
matrices of size
k×k
at most and the crossproduct of at most
|V|
vectors of size
p
. The final product is a
scalar scaling of a
p×p
matrix. Calibrating the clique tree or cluster graph is more complex, because each BP update
has complexity up to
O(k3p3)
. If the phylogeny is a tree, a clique tree has
k= 2
and
|V|= 2n−1
, so that SM-11 has
complexity linear in the number of tips. While PIC can get these estimates in only one traversal of the tree, this formula
requires two traversals of the clique tree, but is more general as it applies to any phylogenetic network.
D.3.4 ML and REML estimation
Restricted maximum likelihood (REML) estimation can be framed as integrating out fixed effects [Harville, 1974], here
µρ
, to estimate covariance parameters, here the BM variance rate
Σ
. This model corresponds to placing an improper
prior on the root using
ℓ(ρ) = +∞
, in which case
µρ
is irrelevant. Then
(SM-11)
remains valid (with vanishing terms
for the root) and gives an analytical formula for the REML estimate of Σ.
39

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
For maximum likelihood (ML) estimation of
µρ
, considered as the state
Xρ
at the root node
ρ
, we need to consider
the case
ℓ(ρ)=0
to fix
Xρ=µρ
. Under
ℓ(ρ)=0
,
(SM-10)
cannot be calculated because
Xρ=µρ
was absorbed as
evidence and
ρ
removed from scope. Instead, we note that under an improper root with infinite variance, the posterior
density of the root trait conditional on all the tips is proportional to the likelihood
p(Xρ|Y)∝p(Y|Xρ)×p(Xρ)
because
p(Xρ)≡1
under an improper prior on
Xρ
. Therefore, maximizing the likelihood
p(Y|Xρ)
in the root
parameter
Xρ=µρ
amounts to maximizing the density
p(Xρ|Y)
in
Xρ
. This density is Gaussian with expectation
Eρ
by
(SM-4)
so its maximum is attained at
ˆµρ= Eθ[Xv|Y] = Eρ
. In summary, the ML estimate of
µρ
is still given
by (SM-10), but calculated by running BP under an improper prior at the root.
D.4 Analytical formula for phylogenetic regression
In the previous section, we derived analytical formulas
(SM-10)
and
(SM-11)
for estimating the parameters of a
homogeneous multivariate BM on a phylogenetic network, using the output of only one BP calibration thanks to
Corollary 4.
Instead of fitting a multivariate process, it is often of interest to look at the distribution of one particular trait conditional
on all others. This phylogenetic regression setting is for instance used on a network in Bastide et al. [2018b]. Writing
V
the (univariate) trait of interest measured at the
n
tips of a network, and
U
the
p×n
matrix of regressors, we are
interested in the model:
V=U⊤β+ϵ, (SM-12)
with
β
a vector of
p
coefficients, and
ϵ
a vector of residuals with expectation
0
and a variance-covariance matrix that is
given by a univariate BM on the network with variance rate
σ2
, and a root fixed to
0
. The
vth
column
U•v
corresponds
to the predictors at leaf vand will be denoted as Uv.
In this setting, explicit maximum likelihood estimators for
β
and
σ
are available, but they involve the inverse of an
n×nmatrix, with O(n3)complexity. Our goal is to get these estimators in linear time.
D.4.1 Parameter estimation using the joint distribution
To build on section D.3, we first look at the joint distribution of the reponse and predictors
V
and
U
. Setting the
intercept aside, we slightly rewrite model SM-12 (with a slight change of notation for Uand p) to:
V=α1+U⊤β+ϵ, (SM-13)
with
α
a scalar,
1
the vector of ones,
β
a vector of
p
coefficients, and
ϵ
a vector of residuals with expectation
0
and
a variance-covariance matrix given by a univariate BM on the network with variance rate
σ2
, and a root fixed to
0
.
Assuming that the joint trait
X= (V, U )
, of dimension
p+ 1
, is jointly Gaussian and evolving on the network with
variance rate ΣX=ΣV V ΣV U
ΣUV ΣU U , we obtain the regression model above with
β=Σ−1
UU ΣU V and σ2=ΣV V −ΣV U Σ−1
UU ΣU V .(SM-14)
This is because a joint BM evolution for
X
implies that the evolutionary changes in
V
and
U
along each branch
e
,
(∆V)e
and
(∆U)e
, are jointly Gaussian
N(0, ℓ(e)ΣX)
and independent of previous evolutionary changes. By classical
Gaussian conditioning, this means that
(∆V)e= (∆U)⊤
eβ+ (∆ϵ)e
where
(∆ϵ)e∼ N(0, ℓ(e)σ2)
and independent of
(∆U)e
. At a hybrid node, the merging rule holds for both
V
and
U
with the same inheritance weights, so by induction on the nodes (in preorder) we get that
Vu=α+U⊤
uβ+ϵu
at
every node
u
in the network, with
α=Vρ−U⊤
ρβ
, and with
ϵ
following a BM process with variance rate
σ2
starting at
ϵρ= 0. Therefore SM-13 holds at the tips.
Consequently, we can apply formulas
(SM-10)
and
(SM-11)
to get maximum likelihood (or REML) estimates
ˆµX
and
ˆ
ΣXof the joint expectation and variance rate matrix of X. We can then plug in these estimates in SM-14 to get:
ˆα= ˆµV−ˆ
ΣV U ˆ
Σ−1
UU ˆµU,ˆ
β=ˆ
Σ−1
UU ˆ
ΣUV ,and ˆσ2=ˆ
ΣV V −ˆ
ΣV U ˆ
Σ−1
UU ˆ
ΣUV ,(SM-15)
where
ˆµV
and
ˆµU
are, respectively, the scalar and vector of size
p
extracted from
ˆµX
for traits
V
and
U
, and, similarly,
ˆ
ΣV U
,
ˆ
ΣUV
,
ˆ
ΣV V
and
ˆ
ΣUU
are the sub-matrices of dimension
1×p
,
p×1
,
1×1
and
p×p
extracted from
ˆ
ΣX
. As
calculating
ˆµX
and
ˆ
ΣX
via
(SM-10)
and
(SM-11)
has complexity
O(|V|(k3+p3))
where
|V|
is the number of nodes
in the network and
k
is the maximum cluster size, obtaining
ˆα
,
ˆ
β
and
ˆσ2
with SM-15 has that same complexity, which
can be much smaller than O(n3). If the phylogeny is a tree, this complexity depends linearly on n.
40

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
D.4.2 Direct parameter estimation using the marginal distribution
Going back to model (SM-12), we do not assume that X= (V, U )is jointly Gaussian and make no assumption about
U
. The distribution assumption is solely on the residual
ϵ
. Model SM-12 then amounts to a trait
Yβ=V−U⊤β
at
the tips (for a given
β
) evolving under a homogeneous univariate BM model with variance
σ2
. We denote by
X
the
corresponding trait at all network nodes, whose values Yβat tips depends on β.
We can apply Fisher’s formula (6.1) to this model, taking the derivative with respect to β:
∇β′log p(V−U⊤β′)β′=β=X
v∈V
Eθh∇β′log ϕv(Xv|Xpa(v), θ′)β′=βV−U⊤βi(SM-16)
In this sum, the only factors
ϕv
that depend on
β′
are the factors at the tips. In phylogenies, leaves are typically con-
strained to have a single parent, although extending our derivation to the case of hybrid leaves would be straightforward.
For a leaf
v
with parent
pa(v) = {u}
, we have:
Yβ′
v|Xpa(v)∼ NXu;σ2ℓ(v)
, and
Yβ′
v=Vv−U⊤
vβ′
, so that
ϕv(Yβ′
v|Xpa(v), β′) = ϕv(Vv|U⊤
vβ′+Xu, β′). Using the Gaussian derivative formula (SM-9), we get:
∇β′hlog ϕv(Yβ′
v|Xpa(v), θ′)iβ′=β=∇β′log ϕv(Vv|U⊤
vβ′+Xu, θ′)β′=β
=∂(Xu+U⊤
vβ′)
∂β′
⊤
β′=β
[σ2ℓ(v)]−1Vv−Xu+U⊤
vβ
=Uv[σ2ℓ(v)]−1Vv−U⊤
vβ−Xu,
Using Lemma 3, the expectation of
Xu
conditional on the observed values at the tips,
Eθ(XuYβ)
, is linear in the
data so that by (SM-4):
Eθ(XuYβ)=Eθ(XuYβ)⊤= [J−1
u]u•(MV
u)⊤−[J−1
u]u•(MU
u)⊤β=EV
u−(EU
u)⊤β,
where
MV
u
,
EV
u
and
MU
u
,
EU
u
denote, respectively, the BP quantities of Lemma 3 when applied to the traits
V
and
U
separately. Note that
MU
u
can also be obtained by running BP on each of the
p
rows of
U
independently, because
Ju
does not depend on the data and
Mu
depends linearly on the data. As
V
is a trait of dimension
1
,
MV
u
is a row
vector of size
s
, the number of nodes in the chosen cluster containing
u
; and
[J−1
u]u•(MV
u)⊤=EV
u
is a scalar. Also,
[J−1
u]u•(MU
u)⊤= (EU
u)⊤
is a row-matrix of size
1×p
, so that
[J−1
u]u•(MU
u)⊤β= (EU
u)⊤β
is also a scalar. We can
hence write, for leaf vwith parent u:
Eθh∇β′[log ϕv(Xv|Xu, θ)]|β′=βi=Uv[σ2ℓ(v)]−1Vv−U⊤
vβ−(EV
u−(EU
u)⊤β).
Taking the sum and cancelling the gradient in β, we get:
b
β= X
leaf v
1
ℓ(v)Uv(Uv−EU
pa(v))⊤!−1X
leaf v
1
ℓ(v)Uv(Vv−EV
pa(v)).(SM-17)
Note that the first term of the product involves the inversion of a
p×p
matrix, and that this formula outputs a vector
of size
p
. To get all the quantities needed in this formula, we just need one BP calibration of the cluster graph with
multivariate traits
(V, U )
to get the conditional means and variances, which can be done efficiently using only univariate
traits thanks to Corollary 5.
Finally, to get an estimator of the residual variance
σ2
, we can run another BP calibration, taking
ˆϵ=V−U⊤b
β
as the
tip trait values, and then use the formulas from the previous section. Using an infinite root variance for this last BP
traversal gives us the REML estimate of the variance.
If the phylogeny is a tree, this algorithm involves
p+ 2
univariate BP calibrations, each requiring two traversals of the
tree, sums of
O(n)
terms in SM-17 and other formulas, and a
p×p
matrix inversion, so calculating
b
β
and
ˆσ2
is linear
in the number of tips. Comparatively, the algorithm used in the
R
package
phylolm
[Ho and An
´
e, 2014] only needs one
multivariate traversal of the tree. Our algorithm is more general however, as it applies to any phylogenetic network and
to any associated cluster graph.
E Regularizing initial beliefs
At initialization, each factor is assigned to a cluster whose scope includes all nodes from that factor. Then the initial
belief
βi
of a cluster
Ci
is the product of all factors assigned to it by (4.2). Sepsets are not assigned any factors so their
41

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
beliefs
µi,j
are initialized to 1. This assignment guarantees that the the joint density
pθ
of the graphical model equals
the following quantity at initialization:
QCi∈V∗βi
Q{Ci,Cj}∈E∗µi,j
.(SM-18)
(SM-18)
is called the graph invariant because BP modifies cluster and sepset beliefs without changing the value of this
quantity, and hence keeps it equal to
pθ
[Koller and Friedman, 2009]. Initialization with (4.2) can lead to degenerate
messages, as highlighted in section 7.1. However, other belief assignments are permitted, provided that
(SM-18)
equals
pθat initialization. Modifying beliefs between BP iterations is also permitted, provided that (SM-18) is unchanged.
Regularization modifies the belief precisions to make them non-degenerate. To maintain the graph invariant, every
modification to a cluster belief is balanced by a modification to an adjacent sepset belief. We describe two basic
regularization algorithms below, but many others could also be considered.
Algorithm R3 Regularization along variable subtrees
1: for all variable xdo
2: Tx←subtree induced by all clusters containing x
3: fix ϵ > 0
4: for all sepsets and for all but one cluster in Txdo
5: add ϵto the diagonal entry of its belief’s precision matrix corresponding to x
Algorithm R4 Regularization on a schedule
1: Choose an ordering of clusters: C1,...,C|V |
2: For each cluster Ciand each neighbor Cjof Ci, set i→jas unvisited
3: for all i= 1,...,|V| do
4: for all neighbor Cjof Cido
5: if j→iis unvisited then
6: fix ϵ > 0
7: add ϵIto the precision matrix of the sepset Si,j
8: add ϵto the diagonal entry of Ci’s precision matrix corresponding to each variable in Si,j
9: mark j→ias visited
10: for all neighbor Ckof Cido
11: if i→kis unvisited then
12: propagate belief from Cito Ckby Algorithm 2
13: mark i→kas visited
In Algorithm R3, each modified belief is multiplied by a regularization factor
exp −1
2ϵx2
. The graph invariant is
satisfied because
Tx
must be a tree (by the running intersection property), so the same number of clusters and sepsets
are modified and the regularization factors cancel out in
(SM-18)
. In Algorithm R4, the same argument applies to
modifications on lines 7 and 8, which cancel out in
(SM-18)
so the graph invariant is maintained. It is also maintained
on line 12, which uses BP.
The choice of the regularization constant
ϵ
is not specified above, but should be adapted to the magnitude of entries in
the affected precision matrices.
Both algorithms performed comparably well on the join-graph structuring cluster graphs used in Fig. 7 and Fig. S2. On
the factor graph for the complex network, however, Algorithm R4 was found to work better than R3. Namely, beliefs
remained persistently degenerate after initial regularization with R3, such that the estimated conditional means and
factored energy could not be computed.
Both algorithms are illustrated in Figure S4.
42

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
x6
x8
x12


1−1/2−1/2
−1/2 1/4+3˜ϵ−ϵ1/4
−1/2 1/4 1/4


x10
x11
1+3˜ϵ−ϵ−1
−1 1 x8
x10
1+2ϵ−1
−1 1+ϵ
x5
x8
1−1
−1 1+3˜ϵ−ϵ
x6
x8
x12


1−1/2−1/2
−1/2 1/4+ϵ1/4
−1/2 1/4 1/4


x8
x10
1+ϵ−1
−1 1 
x5
x8
1−1
−1 1 
x10
0+ϵ+ 3˜ϵ−ϵ
x8
0+ϵ+ 3˜ϵ−ϵ
x8
0+ϵ+ 3˜ϵ−ϵ
x8
0+ϵ
x8
0+ϵ
Figure S4: Applying algorithms R3 and R4 on the cluster graph from Fig. 4(d), for a univariate BM model with mean 0
and variance rate 1, edge lengths of 1 in the original network and inheritance probabilities of 0.5. Cluster/sepset
precision matrices have rows labelled by variables to show the nodes in scope. Precision matrices show entries before
regularization (black) and after one pass through the outermost loop of the algorithm (coloured adjustments). Left:
regularization R3 starting with variable
x8
. Right: regularization R4 starting with cluster
{8,10}
, assuming that it is the
first cluster scheduled to be processed. For R4, we differentiate the effects of lines 3-8 (blue) and lines 9-12 (red). For
example, the resulting precision matrix for sepset {x10}is [3˜ϵ]after summing these effects, where ˜ϵ=ϵ+o(ϵ).
F Handling deterministic factors
This section illustrates two approaches to running BP in the presence of a deterministic Gaussian factor that arises
because the state at a hybrid node is a linear combination of its parents’ states.
Let
X
be a univariate continuous trait evolving on the 3-taxon network in Fig. 2a (reproduced in Fig. S5a) under a BM
model with ancestral state 0 at the root and variance rate
σ2
. For simplicity, we assume that tree edges have length 1,
hybrid edges have length 0, and inheritance probabilities are 1/2. The conditional distribution for each node given its
parents is non-deterministic and can be expressed in a canonical form, except for
X5
at the hybrid node. Because of
0-length hybrid edges, we have the deterministic relationship:
X5= (X4+X6)/2
. After absorbing evidence
x1,x2,x3
at the tips and fixing xρ= 0, the factors are:
ϕ1=C(x4;σ−2, σ−2x1, g1)ϕ2=C(x5;σ−2, σ−2x2, g2)ϕ3=C(x6;σ−2, σ −2x3, g3)
ϕ4=C(x4;σ−2,0, g4)ϕ6=C(x6;σ−2,0, g6)ϕ5=δ(x5−(x4+x6)/2)
where
gi
normalizes
ϕi
to a valid probability density and
δ(·)
denotes a Dirac distribution at 0.
U
in Fig. 2b (reproduced
in Fig. S5b) remains a valid clique tree for this model. We index the cliques in
U
as
C1={x1, x4}
,
C2={x2, x5}
,
C3={x3, x6}
,
C4={x5, x4, x6}
,
C5={x4, x6, xρ}
. We set
C5
as the root clique and assume the following factor
assignment for U:ϕ17→ C1,ϕ27→ C2,ϕ37→ C3,ϕ57→ C4,{ϕ4, ϕ6} 7→ C5.
F.1 Substitution
The substitution approach removes the Dirac factor
ϕ5
by removing
x5
from the model, substituting it by
(x4+x6)/2
where needed. Since ϕ2has scope {x5}, it is reparametrized as ϕ′
2on scope {x4, x6}:
ϕ′
2=Cx4
x6;1
4σ21 1
1 1,x2
2σ21
1, g2.
For the simple univariate BM, it is well known in the admixture graph literature [Pickrell and Pritchard, 2012] that this
substitution corresponds to using a modified network
N′
in which hybrid edges do not all have length 0 (Fig. S5c).
N′
is built from the original network
N
by removing the hybrid node 5 and connecting its parents (nodes 4 and 6)
to its child (node 2) with edges of lengths
ℓ4=ℓ6= 2
for example (to ensure that
γ2
4ℓ4+γ2
2ℓ2
equals the length
of the original child edge to node 2). A clique tree
U′
for
N′
can be obtained from
U
by replacing
C2
and
C4
with
C′
4={x2, x4, x6}(Fig. S5d). Factor ϕ′
2is assigned to C′
4while the other factor assignments stay the same.
43

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
N
(a)
xρ
x4x5x6
x1x2x3
U
(b)
x4, x6, xρ
C5
x2, x4, x6
C4
x1, x4
C1x2, x5
C2x3, x6
C3
N′
(c)
xρ
x4x6
x1x2x3
U′
(d)
x4, x6, xρ
C5
x2, x4, x6
C4′
x1, x4
C1x3, x6
C3
x4x6
x2
x4, x6
γ= 0.5γ= 0.5
x4x6
x4, x6
Figure S5: (a) Network
N
from Fig. 2a. (b) Clique tree
U
from Fig. 2b. (c) Network
N′
obtained by removing the
hybrid node 5 from
N
in (a). The BM model on
N
leads to the same probability model for the nodes in
N′
as the
BM model on
N′
, given a valid assignment of hybrid edge lengths in
N′
(see text). (d) Clique tree
U′
for
N′
after
moralization.
Standard BP can be used for the BM model on
N′
because all factors are non-degenerate. After one postorder traversal
of U′, the message ˜µ4′→5, final belief β5and log-likelihood LL(σ2) = log pσ2(x1,x2,x3)are:
˜µ4′→5=ψ4′˜µ1→4′˜µ3→4′=ϕ′
2ϕ1ϕ3=C x4
x6;1
4σ25 1
1 5,1
2σ22x1+ x2
2x3+ x2,
3
X
i=1
gi!
β5=ψ5˜µ4′→5=ϕ4ϕ6˜µ4′→5=C
x4
x6;K=1
4σ29 1
1 9, h =1
2σ22x1+ x2
2x3+ x2, g =
6
X
i=1,i=5
gi

LL(σ2) = Zβ5dx4dx6=
6
X
i=1,i=5
gi+

log 2π1
4σ29 1
1 9−1+



1
2σ22x1+ x2
2x3+ x2


 1
4σ2"9 1
1 9#!−1

/2
We can still recover the conditional distribution of
X5
from
β5
, because
β5
has scope
{x4, x6}
. Let
(K, h, g)
be the
parameters of the canonical form of
β5
, given above. After the postorder traversal of
U′
,
β5
contains information
from all the tips such that the distribution of
(X4, X6)
conditional on the data
(x1,x2,x3)
is
NK−1h, K−1
. Since
X5=γ⊤X4
X6with γ⊤= [1/2,1/2], we get
X5|(x1,x2,x3)∼ N(γ⊤K−1h, γ ⊤K−1γ) = N 3
X
i=1
xi/5, σ2/5!.
F.2 Generalized canonical form
A more general approach generalizes canonical form operations to include Dirac distributions without modifying the
original set of factors and clique tree, as demonstrated in Schoeman et al. [2022]. Crucially, they derived message
passing operations (evidence absorption, marginalization, factor product, etc.) for a generalized canonical form:
D(x;Q,R,Λ, h, c, g):=C(Q⊤x;Λ, h, g)·δ(R⊤x−c)
where
x
is an
n
-dimensional vector,
Λ⪰0
is a
(n−k)×(n−k)
diagonal matrix, and
Q
and
R
are matrices of
dimension
n×(n−k)
and
n×k
respectively, that are orthonormal and orthogonal to each other, that is:
Q⊤Q=In−k
,
R⊤R=Ik
, and
Q⊤R=0
. If
Q
is square (thus invertible) then
R
is empty and the Dirac
δ(·)
term is dropped or
defined as 1. The same applies to the
C(·)
term if
R
is square (and
Q
is empty). Non-deterministic linear Gaussian
44

SM: Leveraging graphical model techniques to study evolution on phylogenetic networks
factors are represented in generalized canonical form with
Q
square from the eigendecomposition of
K=QΛQ⊤
.
Thus, we can run BP on U, converting beliefs or messages to generalized canonical form as needed.
Running BP according to a postorder traversal of U, the message ˜µ4→5involves a degenerate component:
˜µ4→5=Zψ4
3
Y
i=1
˜µi→4dx5=Zϕ5
3
Y
i=1
ϕidx5.
To compute ˜µ4→5, we first convert ϕ5and Q3
i=1 ϕito generalized canonical forms:
ϕ5=D "x4
x6
x5#;"1/w11/w2
1/w1−1/w2
1/w10#,"1/2w0
1/2w0
−1/w0#,Λ5=0 0
0 0,0
0,0,0!
3
Y
i=1
ϕi=D "x4
x6
x5#;I3,−, σ−2I3, σ−2"x1
x3
x2#,−,
3
X
i=1
gi!
where
(w0, w1, w2)=(p6/4,√3,√2)
are normalization constants for the respective columns, and the dashes indicate
that the δ(·)part is dropped. By Schoeman et al. [2022, Algorithm 3], their product evaluates to:
ϕ5
3
Y
i=1
ϕi=D "x4
x6
x5#;Q="1/w11/w2
1/w1−1/w2
1/w10#,R="1/2w0
1/2w0
−1/w0#, σ−2I2, h =σ−2(x1+ x2+ x3)/w1
(x1−x3)/w2,0,
3
X
i=1
gi!
We partition Q=Q4,6
Q5and R=R4,6
R5by separating the first two rows from the last row.
Then by Schoeman et al. [2022, Algorithm 2], integrating out x5yields:
ZD "x4
x6
x5#;Q4,6
Q5,R4,6
R5, σ−2I2, h, 0,
3
X
i=1
gi!dx5=D x4
x6;Q4→5,−,Λ4→5, h4→5,−,
3
X
i=1
gi!
where, following notations from Schoeman et al. [2022] for intermediate quantities in their Algorithm 3:
U=I2,W= [1]
F= (W(R5W)+Q5)⊤=−w0/w1
0
G= (Q⊤
4,6−F R⊤
4,6)U=3/(2w1) 3/(2w1)
1/w2−1/w2
ZΛ4→5Z⊤=SVD(G⊤(σ−2I2)G) = SVD 1
4σ25 1
1 5
therefore Z=1/w21/w2
1/w2−1/w2and Λ4→5=1
2σ23 0
0 2
Q4→5=U Z =Z
h4→5=Z⊤G⊤h=σ−2w2(x1+ x2+ x3)
w2(x1−x3)
This generalized canonical form can be rewritten as a standard canonical form:
˜µ4→5=C Q⊤
4→5x4
x6;Λ4→5, h4→5,
3
X
i=1
gi!=C x4
x6;Q4→5Λ4→5Q⊤
4→5,Q4→5h4→5,
3
X
i=1
gi!
=C x4
x6;1
4σ25 1
1 5,1
2σ22x1+ x2
2x3+ x2,
3
X
i=1
gi!
which agrees with
˜µ4′→5
from the substitution approach in F.1. Since the remaining operations to compute
β5=
ψ5˜µ4→5
and
LL(σ2) = Rβ5dx4dx6
involve non-deterministic canonical forms, it is clear that they both evaluate to the
same quantity as when using the substitution approach above.
45