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Inference via low-dimensional couplings

Banff International Research Station for Mathematical Innovation and Discovery

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 filtering, smoothing, and online 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. This is joint work with Alessio Spantini and Daniele Bigoni.

Mathematics; Calculus of variations and optimal control; optimization; Statistics; Scientific computing

video/mp4

Author affiliation: Massachusetts Institute of Technology

2017-10-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63470

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/