calculations and simulation, and running under Macintosh OSX version 10.9.5 on a 2.5 GHz Intel Core i7 processor. We tested the automated blocking procedure on seven examples, including real-data and toy models, and compared the results against standard MCMC algorithms. When there are obvious “hand- tuned” algorithms that a seasoned MCMC practitioner would consider for a particular model, we included those as well. For the toy models, the goal was to construct posterior distributions with specific correlation structures as described below. In these cases the models are simply prior distributions without any likelihood component. This model structure demonstrates parameter groups of varying size, where each group exhibits a fixed intra-group pairwise correlation. The model contains N = 64 parameters, half of which are grouped to have pairwise posterior correlation of ρ . This is accomplished using a prior multivariate normal distribution with appropriate covariance (equivalently, correlation) matrix, which in the absence of a likelihood term fully determines the posterior distribution for these 32 parameters. Similarly, additional disjoint groups of correlated parameters are constructed of sizes 16, 8, 4, and 2. The remaining two parameters are uncorrelated to any others, specified using univariate normal prior distributions. We consider three values for the intra-group correlation, ρ = 0.2, 0.5, and 0.8. As these models have no likelihood, we are using MCMC to sample from the prior distribution. We note that the dependence structure is the same as the block-diagonal covariance structure (with the blocks having compound symmetry) obtained when analytically integrating over the exchangeable prior means of clustered random effects. This example thereby mimics the structure found in basic multilevel hierarchical models, albeit without the explicit computational expense of a likelihood calculation. 18 Varying Size Blocks of Fixed Correlation 4.2 Model Descriptions