[Paper Review] Selection of proposal distributions for generalized importance sampling estimators

Vivekananda Roy, Evangelos Evangelou|arXiv (Cornell University)|May 2, 2018

Statistical Methods and Bayesian Inference40 references3 citations

TL;DR

This paper proposes three methods for selecting proposal distributions in generalized importance sampling estimators to improve stability and efficiency in multi-target inference. It introduces a geometric coverage criterion using symmetric KL divergence, a minimax variance approach, and a maximum entropy method—each enhancing estimator performance through variance reduction and coverage optimization, with consistent spectral variance estimators provided for key estimators.

ABSTRACT

The standard 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 generalized IS estimators involving samples from multiple proposal distributions are preferred. Just like the standard 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 based on (i) a geometric space filling coverage criterion, (ii) a minimax variance approach, and (iii) a maximum entropy approach. The first two methods are applicable to any multi-proposal 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 coverage 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 estimator, respectively. Thus, we provide consistent spectral variance estimators for these asymptotic variances. The proposed methods for selecting proposal densities are illustrated using various detailed examples.

Motivation & Objective

• To address the instability of standard importance sampling in multi-target inference due to high-variance weights.
• To develop systematic methods for selecting multiple proposal distributions that enhance estimator performance.
• To ensure robustness and efficiency in generalized importance sampling by optimizing proposal selection.
• To provide consistent estimators for asymptotic variances in Doss's and Geyer's importance sampling estimators.
• To demonstrate the practical effectiveness of the proposed selection methods through detailed numerical examples.

Proposed method

• Proposes a geometric space-filling coverage criterion based on symmetric Kullback-Leibler divergence to measure proposal distribution spread.
• Introduces a minimax variance approach that selects proposals to minimize the asymptotic variance of Doss's (2010) generalized IS estimator.
• Develops a maximum entropy approach tailored for Doss's two-stage IS estimator, promoting diversity and coverage of the target space.
• Derives consistent spectral variance estimators for the asymptotic variances of Doss's and Geyer's (1994) reverse logistic IS estimators.
• Uses these variance estimates to guide proposal selection, ensuring improved estimator stability and efficiency.
• Employs numerical examples to validate the performance of the proposed selection criteria across different scenarios.

Experimental results

Research questions

• RQ1 How can proposal distributions be selected to minimize the asymptotic variance in generalized importance sampling with multiple proposals?
• RQ2 What is an effective measure of coverage for multiple proposal distributions that ensures robust sampling across the target space?
• RQ3 Can a maximum entropy criterion improve the performance of two-stage generalized IS estimators?
• RQ4 How can consistent spectral estimators of asymptotic variance be constructed for Doss's and Geyer's IS estimators?
• RQ5 To what extent do the proposed selection methods improve estimator stability and accuracy in multi-target inference?

Key findings

• The symmetric Kullback-Leibler divergence provides a reliable measure of coverage for multiple proposal distributions, enabling effective space-filling selection.
• The minimax variance approach significantly reduces the asymptotic variance of Doss's generalized IS estimator by optimizing proposal distribution configurations.
• The maximum entropy method enhances estimator performance in Doss's two-stage IS framework by promoting diversity and reducing concentration of samples.
• Consistent spectral estimators for asymptotic variances are successfully derived for both Doss's and Geyer's IS estimators, enabling reliable variance-based optimization.
• Numerical examples demonstrate that the proposed methods lead to more stable and efficient estimators compared to naive or uniform proposal selection.
• The combination of coverage, variance minimization, and entropy maximization leads to robust and scalable proposal selection in complex multi-target inference problems.