Estimation

MCMC estimation was performed for the diffusion tensor model and for the simple partial volume model. In both cases parameters were initialized with a least squares diffusion tensor fit. The Markov Chains were then jumped 500 times without sampling as a ``burnin'' (see [12]), followed by 2000 times, sampling every second jump, to give 1000 samples. A single jump of the parameter set consisted of independent jumps of each parameter. In both models samples were drawn from the precision ( $\frac{1}{\sigma^2}$) with a Gibbs sampler, and from all other parameters with Metropolis Hastings samplers. Proposal distributions for Metropolis Hastings parameters were zero mean Gaussians with standard deviations tuned adaptively to give a jump acceptance rate of 0.5. The full conditional distributions for the Gibbs sampling of the precision in both models are given in the appendix. Computation time for diffusion data with 63 acquisitions is approximately 0.3 seconds per voxel on a Pentium IV 2GHz. Voxels are processed independently, so computation is easily parallelized.