Selection of proposal distributions for multiple importance sampling

TL;DR

This paper introduces three novel methods for selecting proposal distributions in multiple importance sampling to improve estimator stability and accuracy, especially when using Markov chain samples.

Contribution

It proposes three new approaches—geometric space filling, minimax variance, and maximum entropy—for choosing proposal distributions in multiple IS, with theoretical and practical insights.

Findings

The methods improve estimator stability in multiple IS scenarios.

Spectral variance estimators are provided for Markov chain samples.

Illustrative examples demonstrate the effectiveness of the proposed methods.

Abstract

The naive importance sampling (IS) estimator generally does not work well in examples involving simultaneous inference on several targets, as the importance weights can take arbitrarily large values, making the estimator highly unstable. In such situations, alternative multiple IS estimators involving samples from multiple proposal distributions are preferred. Just like the naive IS, the success of these multiple IS estimators crucially depends on the choice of the proposal distributions. The selection of these proposal distributions is the focus of this article. We propose three methods: (i) a geometric space filling approach, (ii) a minimax variance approach, and (iii) a maximum entropy approach. The first two methods are applicable to any IS estimator, whereas the third approach is described in the context of Doss's (2010) two-stage IS estimator. For the first method, we propose a suitable measure of 'closeness' based on the symmetric Kullback-Leibler divergence, while the second and third approaches use estimates of asymptotic variances of Doss's (2010) IS estimator and Geyer's (1994) reverse logistic regression estimator, respectively. Thus, when samples from the proposal distributions are obtained by running Markov chains, we provide consistent spectral variance estimators for these asymptotic variances. The proposed methods for selecting proposal densities are illustrated using various detailed examples.