Integration against an intractable probability measure is among the fundamental challenges of Bayesian inference. A useful approach to this problem seeks a deterministic coupling of the measure of interest with a tractable "reference" measure (e.g., a standard Gaussian). This coupling is induced by a transport map, and enables direct simulation from the desired measure simply by evaluating the transport map at samples from the reference. Approximate transports can also be used to "precondition" standard Monte Carlo schemes. Yet characterizing a suitable transport map---e.g., representing, constructing, and evaluating it---grows challenging in high dimensions.
We establish links between the conditional independence structure of the target measure and the existence of certain low-dimensional couplings, induced by transport maps that are sparse or decomposable. Our analysis not only facilitates the construction of couplings in high-dimensional settings, but also suggests certain inference methodologies. For instance, in the context of nonlinear and non-Gaussian state space models, we will describe new variational algorithms for nonlinear smoothing and sequential parameter estimation. We will also outline a new class of nonlinear filters induced by local couplings, for inference in high-dimensional spatiotemporal processes with chaotic dynamics.
This is joint work with Alessio Spantini and Daniele Bigoni.