Session A6 - Mathematical Foundations of Data Assimilation and Inverse Problems - Semi-plenary talk
July 11, 17:00 ~ 18:00 - Room T1
Bayesian inference via low-dimensional couplings
Massachusetts Institute of Technology, USA - email@example.com
Integration against an intractable probability measure is among the fundamental challenges of statistical inference, particularly in the Bayesian setting. A principled 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. Yet characterizing such a map---e.g., representing, constructing, and evaluating it---grows challenging in high dimensions.
We will present 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 describe new variational algorithms for online filtering, smoothing, and parameter estimation. These algorithms implicitly characterize---via a transport map---the full posterior distribution of the sequential inference problem using only local operations while avoiding importance sampling or resampling.
Joint work with Alessio Spantini (Massachusetts Institute of Technology, USA) and Daniele Bigoni (Massachusetts Institute of Technology, USA).