Transport maps for Bayesian computation

Abstract: 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. We also describe conditions, common in Bayesian inverse problems, under which transport maps have a particular low-rank structure. Our analysis not only facilitates the construction of couplings in high-dimensional settings, but also suggests new 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 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.

Biography: Youssef Marzouk is an associate professor in the Department of Aeronautics and Astronautics at the Massachusetts Institute of Technology (MIT), and Director of MIT’s Aerospace Computational Design Laboratory. He is also co-director of educational programs for the MIT Center for Computational Engineering. His research interests lie at the intersection of physical modeling with statistics and computation. He is particularly motivated by uncertainty quantification problems in engineering and geophysical applications. He received his SB, SM, and PhD degrees from MIT and spent several years at Sandia National Laboratories before joining the MIT faculty in 2009. He is a recipient of the Hertz Foundation Doctoral Thesis Prize (2004), the Sandia Laboratories Truman Fellowship (2004-2007), the US Department of Energy Early Career Research Award (2010), and the Junior Bose Award for Teaching Excellence from the MIT School of Engineering (2012). He is an Associate Fellow of the AIAA and serves on the editorial boards of several journals, including the SIAM Journal on Scientific Computing, Advances in Computational Mathematics, and the SIAM/ASA Journal on Uncertainty Quantification.