Bayesian inference with optimal maps

By Tarek A. El-Moselhy, Youssef M. Marzouk. 2011

311 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

A unified framework for multilevel uncertainty quantification in Bayesian inverse problems (2016, unknown): “It will also be interesting to study the applicability and performance of optimal transportation approaches [93, 94] to classical Bayesian inference in the context of multilevel estimation.”
Optimal Mass Transport: Signal processing and machine-learning applications (2017, methodology): “Bayesian inference Another interesting and emerging application of the optimal transport problem is in Bayesian inference [17].”
Bayesian Learning with Wasserstein Barycenters (2018, methodology): “We make the practical assumption that we are capable of generating independent models mi from the posteriors Πn (in the parametric setting, this can be done through efficient Markov Chain Monte Carlo (MCMC) techniques [5, 21, 10] or transport sampling procedures [17, 34, 24, 30]).”
An introduction to sampling via measure transport (2016, methodology): “For the computation of general non-triangular transports see [61]; for a generalization of the framework to compositions of maps see [61, 66]; and for the computation of optimal transports see, for instance, [88, 10, 4, 50].”