Illustration of the (1) Split, Merge and (2) Alter Position moves 

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... the simulations presented below, for the likelihood function we use a value of t = 2 . 57 based on studies of brain contours. Note that this probability model is being setup to assume that there are local parts of the contour which have fractal dimension t . Of course, if it were a fractal and not discretely sampled then by definition this assumption would hold at arbitrary scales. Markov chain simulation (or MCMC) is a method based on drawing values of parameters Θ from functions that approximates it, and then draws are altered to obtain a better approximation to the target posterior distribution p ( Θ | X ) . Since the samples are drawn sequentially and the distribution that generates these depends only on the previous value drawn, the set of states form a Markov chain [18]. To ensure the success of the algorithm it is necessary to create a Markov process whose stationary distribution is the specified p ( Θ | X ) and then run the simulation for a sufficient length of time. The Metropolis-Hastings (MH) algorithm can be considered as a generalisation of a class of Markov chain simulation methods for drawing samples from posterior distributions. Given a candidate-generating density, p ( Θ, Θ ) , of moving from state Θ to Θ , it is clear that the stationary distribution is reached when p ( Θ | X ) p ( Θ, Θ ) = p ( Θ | X ) p ( Θ , Θ ) is satisfied for all Θ , Θ . The MH algorithm performs rejection sam- pling by comparing the validity of the two states by forming an acceptance ratio and then, if α = 1 , the move Θ Θ is accepted, otherwise it is accepted with probability α ≥ r ∼ U [0 , 1] . Note that it is easy to see that if the candidate-generating densities are equal the state changes Θ −→ Θ and Θ −→ Θ , then the chain auto- matically moves to a higher probability state. For the contour partitioning, we allow three moves in the state space: splitting a chosen contour partition at a random point; merging a random pair of neighbouring partitions and altering the position of a boundary between neighbouring segments (see figure 1). The simulation is initialised with a random number of partitions (e.g. 12) and equal candidate-generating probabilities: P ( split ) = P ( merge ) = P ( alter ) = 1 , with moves chosen at random. Note that for 2 the split move, the random position is first chosen in the range 1 . . . N , then the (current) segment in which it lies is split at that point. This ensures that larger segments are more likely to be split than smaller ones. The fractal dimension of a curve estimates of how much space it occupies relative to its length. For a planar curve, it can be calculated by the number, N ( δ ) , of area elements, δ , needed to cover the curve. The estimate, then, of the length of the curve is N ( δ ) δ . The Hausdorff-Besicovitch dimension D of a curve is defined as some measure M d ( δ ) for which N ( δ ) δ d abruptly changes from zero to infinity [19]. Since D is often finite, it can be shown that for sufficiently small ...