in terms of algorithmic and computational efficiencies. From these considerations we conclude that the combination of samplers that yields optimal MCMC efficiency (defined as an MCMC algorithm’s rate of generating effectively independent samples) will be model- and platform-specific. Section 3 introduces a procedure for automated blocking of hierarchical model parameters, designed to maximize the resulting MCMC algorithm efficiency. The main idea of this procedure is to iteratively cluster model parameters based upon empirical posterior correlations, then intelligently subdivide the hierarchical clustering tree (a dendrogram) to determine blockings of parameters that result in efficient MCMC sampling. A series of simulated and real data examples given in section 4 demonstrate that this procedure can improve MCMC efficiency many-fold over statically defined MCMC algorithms and can dynamically generate algorithms comparable in performance to model- specific, hand-tuned algorithms. We close with a discussion in section 5. In this section, we first define a space of valid MCMC algorithms. Then, we examine two domi- nant contributors to the efficiency of an MCMC algorithm: the algorithmic capability to produce effectively-independent samples from the target distribution, and the computational demands of the algorithm in generating MCMC samples; these are composed to form our metric of overall MCMC efficiency. Drawing upon existing asymptotic theory of MCMC sampling, the scaling of computational costs of different sampling schemes, and on several illustrative examples, we argue that achieving an optimally efficient MCMC algorithm for a specific model by pure theory is pro- hibitively difficult. That conclusion motivates our approach of computationally optimizing – or at least greatly improving – MCMC performance through automated exploration of a space of valid MCMC algorithms. We assume a given, fixed, hierarchical model M , which may be represented as a directed acyclic graph where vertices represent top-level model parameters, latent states, or fixed observations (data), and edges represent dependencies between these components. Denote the set of all top-level 5 2.1 MCMC Definition 2 MCMC Algorithms: Definitions and Efficiency