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Heat-maps of the estimated covariance matrix BB T + Σ. Panel (a) plots the data generating covariance matrix. Panels (b), (c), and (d) show the covariance matrix estimated using 1500 posterior samples under each prior setup. Note that the colorbar in panel (b) is 100 times larger than others, indicating the much inflated covariance matrix estimate under the SpSL-IBP prior.
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... of the estimated covariance matrix BB T + Σ from three different approaches are presented in Figure 1. The first panel plots the data generating covariance matrix as a reference. ...
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... three priors induce quite different posterior behaviors, among which the posterior estimate under the modified Ghosh-Dunson prior is the closest to the data generating true value. For the MGP prior, posterior estimate of some zeros elements (blue area in panel (d) of Figure 1) are "twisted"-not penalized to values close to the truth 0. The posterior estimate under the SpSL-IBP prior exhibits an interesting "magnitude inflation" phenomenon (the range of the color-bar in panel (b) is 100 times larger than the others), although the relative magnitudes after rescaling look most similar to the true ones. ...
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... likely explanation is that the normal factor model is only weakly identifiable, and thus the posterior distribution is sensitive to the prior on the high-dimensional loading matrix (Figure 1), if not well controlled. The magnitude inflation phenomenon is an expression of the dominating influence of the independent SpSL prior. ...
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... posterior density of 15 zero elements and 15 nonzero elements (estimated by averaging over the conditional posterior densities) are demonstrated in the supplemental figures in Appendix E.3 (Ma and Liu, 2021). We also observe a similar robustness against the tuning parameter choices in priors of Ghosh and Dunson (2009) and Bhattacharya and Dunson (2011) under the √ n-orthonormal factor model. Figure 10 of Appendix E.3 depicts a comparison between setting v 1 = v 2 = 0.5 versus v 1 = v 2 = 3 in the MGP prior for a simulated dataset. ...
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... of the estimated covariance matrix BB T + Σ from three different approaches are presented in Figure 1. The first panel plots the data generating covariance matrix as a reference. ...
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... three priors induce quite different posterior behaviors, among which the posterior estimate under the modified Ghosh-Dunson prior is the closest to the data generating true value. For the MGP prior, posterior estimate of some zeros elements (blue area in panel (d) of Figure 1) are "twisted"-not penalized to values close to the truth 0. The posterior estimate under the SpSL-IBP prior exhibits an interesting "magnitude inflation" phenomenon (the range of the color-bar in panel (b) is 100 times larger than the others), although the relative magnitudes after rescaling look most similar to the true ones. ...
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... likely explanation is that the normal factor model is only weakly identifiable, and thus the posterior distribution is sensitive to the prior on the high-dimensional loading matrix (Figure 1), if not well controlled. The magnitude inflation phenomenon is an expression of the dominating influence of the independent SpSL prior. ...
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... posterior density of 15 zero elements and 15 nonzero elements (estimated by averaging over the conditional posterior densities) are demonstrated in the supplemental figures in Appendix E.3 (Ma and Liu, 2021). We also observe a similar robustness against the tuning parameter choices in priors of Ghosh and Dunson (2009) and Bhattacharya and Dunson (2011) under the √ n-orthonormal factor model. Figure 10 of Appendix E.3 depicts a comparison between setting v 1 = v 2 = 0.5 versus v 1 = v 2 = 3 in the MGP prior for a simulated dataset. ...
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... There is a vibrant recent literature improving upon and expanding the scope of Bayesian factor analysis (Schiavon et al., 2022;De Vito et al., 2021;Frühwirth-Schnatter, 2023;Roy et al., 2021;Ma and Liu, 2022;Bolfarine et al., 2022;Xie et al., 2022). Even with increasingly rich classes of priors and data types, the canonical approach for posterior computation remains Gibbs samplers that iterate between updating latent factors, factor loadings, residual variances, hyperparameters controlling the hierarchical prior, and other model parameters. ...
Bayesian factor analysis is routinely used for dimensionality reduction in modeling of high-dimensional covariance matrices. Factor analytic decompositions express the covariance as a sum of a low rank and diagonal matrix. In practice, Gibbs sampling algorithms are typically used for posterior computation, alternating between updating the latent factors, loadings, and residual variances. In this article, we exploit a blessing of dimensionality to develop a provably accurate pseudo-posterior for the covariance matrix that bypasses the need for Gibbs or other variants of Markov chain Monte Carlo sampling. Our proposed Factor Analysis with BLEssing of dimensionality (FABLE) approach relies on a first-stage singular value decomposition (SVD) to estimate the latent factors, and then defines a jointly conjugate prior for the loadings and residual variances. The accuracy of the resulting pseudo-posterior for the covariance improves with increasing dimensionality. We show that FABLE has excellent performance in high-dimensional covariance matrix estimation, including producing well calibrated credible intervals, both theoretically and through simulation experiments. We also demonstrate the strength of our approach in terms of accurate inference and computational efficiency by applying it to a gene expression data set.
Change-points detection has long been important and active research areas with broad application in economics, medical research, genetics, social science, etc. It includes themes of the number, locations, and jump sizes of change-points. However, most existing methods focus on segment parameters or features, regardless of Bayesian or frequentist. We propose an innovative non-segmental approach to detect multiple change-points by concentrating the abrupt changes into a global distributional parameter, which is characterized by a function of states of the system. We construct a class of discrete spike and Cauchy-slab variate prior, which distinguishes from existing such two-group mixture priors by a dynamic rate of a latent indicator. A 3-sigma discrimination criterion of change-points is built with sigma being the standard deviation of the sequence of differences between marginal maximum a posteriori estimates on two adjacent discretized states. It intrinsically guarantees reasonable false positive rate to prevent over-detection. The proposed method is powerful and robust to unveil structure changes of the autoregression model for house prices in London and the linear regression model for age-specific fertility in US, not to mention consistent detection results of shifts of mean or scale on known data sets. Abundant simulations demonstrate that the proposed method outperforms state-of-the-art others in finite-sample performance. An R package \texttt{BaRDCP} is developed for detecting changes in mean, scale, and the regression or autocorrelation coefficient.
Blind source separation (BSS) aims to separate latent source signals from their mixtures. For spatially dependent signals in high dimensional and large-scale data, such as neuroimaging, most existing BSS methods do not take into account the spatial dependence and the sparsity of the latent source signals. To address these major limitations, we propose a Bayesian spatial blind source separation (BSP-BSS) approach for neuroimaging data analysis. We assume the expectation of the observed images as a linear mixture of multiple sparse and piece-wise smooth latent source signals, for which we construct a new class of Bayesian nonparametric prior models by thresholding Gaussian processes. We assign the vMF priors to mixing coefficients in the model. Under some regularity conditions, we show that the proposed method has several desirable theoretical properties including the large support for the priors, the consistency of joint posterior distribution of the latent source intensity functions and the mixing coefficients, and the selection consistency on the number of latent sources. We use extensive simulation studies and an analysis of the resting-state fMRI data in the Autism Brain Imaging Data Exchange (ABIDE) study to demonstrate that BSP-BSS outperforms the existing method for separating latent brain networks and detecting activated brain activation in the latent sources.
