Sampling via Measure Transport: An Introduction

Youssef Marzouk (), Tarek Moselhy (), Matthew Parno () and Alessio Spantini ()
Additional contact information
Youssef Marzouk: Massachusetts Institute of Technology
Tarek Moselhy: D. E. Shaw Group
Matthew Parno: Massachusetts Institute of Technology
Alessio Spantini: Massachusetts Institute of Technology

Chapter 23 in Handbook of Uncertainty Quantification, 2017, pp 785-825 from Springer

Abstract: Abstract We present the fundamentals of a measure transport approach to sampling. The idea is to construct a deterministic coupling – i.e., a transport map – between a complex “target” probability measure of interest and a simpler reference measure. Given a transport map, one can generate arbitrarily many independent and unweighted samples from the target simply by pushing forward reference samples through the map. If the map is endowed with a triangular structure, one can also easily generate samples from conditionals of the target measure. We consider two different and complementary scenarios: first, when only evaluations of the unnormalized target density are available and, second, when the target distribution is known only through a finite collection of samples. We show that in both settings, the desired transports can be characterized as the solutions of variational problems. We then address practical issues associated with the optimization-based construction of transports: choosing finite-dimensional parameterizations of the map, enforcing monotonicity, quantifying the error of approximate transports, and refining approximate transports by enriching the corresponding approximation spaces. Approximate transports can also be used to “Gaussianize” complex distributions and thus precondition conventional asymptotically exact sampling schemes. We place the measure transport approach in broader context, describing connections with other optimization-based samplers, with inference and density estimation schemes using optimal transport, and with alternative transformation-based approaches to simulation. We also sketch current work aimed at the construction of transport maps in high dimensions, exploiting essential features of the target distribution (e.g., conditional independence, low-rank structure). The approaches and algorithms presented here have direct applications to Bayesian computation and to broader problems of stochastic simulation.

Keywords: Measure transport; Optimal transport; Knothe–Rosenblatt map; Monte Carlo methods; Bayesian inference; Approximate Bayesian computation; Density estimation; Convex optimization (search for similar items in EconPapers)
Date: 2017
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-12385-1_23

Ordering information: This item can be ordered from
http://www.springer.com/9783319123851

DOI: 10.1007/978-3-319-12385-1_23

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().