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MALA-within-Gibbs samplers for high-dimensional distributions with sparse conditional structure

Markov chain Monte Carlo (MCMC) samplers are numerical methods for drawing samples from a given target probability distribution. We discuss one particular MCMC sampler, the MALA-within-Gibbs sampler, from the theoretical and practical perspectives. We first show that the acceptance ratio and step size of this sampler are independent of the overall problem dimension when (i) the target distribution has sparse conditional structure, and (ii) this structure is reflected in the partial updating strategy of MALA-within-Gibbs. If, in addition, the target density is block-wise log-concave, then the sampler's convergence rate is independent of dimension. From a practical perspective, we expect that MALA-within-Gibbs is useful for solving high-dimensional Bayesian inference problems where the posterior exhibits sparse conditional structure at least approximately. In this context, a partitioning of the state that correctly reflects the sparse conditional structure must be found, and we illustrate this process in two numerical examples. We also discuss trade-offs between the block size used for partial updating and computational requirements that may increase with the number of blocks.