Bayesian Inference with Optimal Maps

TL;DR

This paper introduces a novel Bayesian inference method using optimal transport maps to directly transform the prior into the posterior, avoiding Markov chain simulation and enabling efficient, independent sampling.

Contribution

It develops a map-based Bayesian inference framework grounded in optimal transport theory, providing explicit parameterization, efficient computation, and convergence criteria.

Findings

Overcomes computational bottlenecks of MCMC

Provides analytical posterior moments

Enables independent posterior sampling without additional likelihood evaluations

Abstract

We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measure-preserving map is established by formulating the problem in the context of optimal transport theory. We discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem, exploiting gradient information from the forward model when possible. The resulting algorithm overcomes many of the computational bottlenecks associated with Markov chain Monte Carlo. Advantages of a map-based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves. The optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood. We demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension, involving the inference of parameters appearing in ordinary and partial differential equations.