TITLE:  Measure transport approaches for Bayesian computation

ABSTRACT:

We will discuss how transport maps, i.e., deterministic couplings between probability measures, can enable useful new approaches to Bayesian computation. A first use involves a combination of optimal transport and Metropolis correction; here, we use continuous transportation to transform typical MCMC proposals into adapted non-Gaussian proposals, both local and global. Second, we discuss a variational approach to Bayesian inference that constructs a deterministic transport map from a reference distribution to the posterior, without resorting to MCMC. Independent and unweighted samples can then be obtained by pushing forward reference samples through the map.

Making either approach efficient in high dimensions, however, requires identifying and exploiting low-dimensional structure. We present new results relating the sparsity and decomposability of transports to the conditional independence structure of the target distribution. We also describe conditions, common in inverse problems, under which transport maps have a particular low-rank or near-identity structure. In general, these properties of transports can yield more efficient algorithms. As a particular example, we derive new deterministic "online" algorithms for Bayesian inference in nonlinear and non-Gaussian state-space models with static parameters. 

This is joint work with Daniele Bigoni, Matthew Parno, and Alessio Spantini.