Designing Zero-Mean Feature Functions for Multimodal Distributions

2026-06-04

AI Summary

This work addresses the limited variance reduction efficacy of conventional zero-mean control variates constructed via Stein’s identity when applied to multimodal distributions. To overcome this limitation, the authors propose a novel approach that integrates distribution approximation with density ratio estimation to construct zero-mean control variates better suited for multimodal settings. These enhanced variates are then combined with Stein functions within a control variate framework. The proposed strategy effectively mitigates the shortcomings of existing Stein-based methods in complex, multimodal scenarios. Empirical results on bimodal distributions demonstrate a substantial reduction in the variance of Monte Carlo expectation estimates, leading to significantly improved estimation accuracy.

Abstract

To improve the accuracy of Monte Carlo estimation of expectations, a set of zero-mean feature functions, known as control variates, can be used. They can be used as feature functions for linear regression of the target function, and we can obtain an unbiased and variance-reduced estimate using its residual. One known way to construct such functions is a method using an equality called Stein's identity, but these functions are not sufficient for the case where the target distribution is multimodal. We propose a different approach to constructing these zero-mean functions based on distribution approximation and the density ratio. We demonstrate that combining the functions constructed by these two strategies can effectively reduce the estimation variance for a bimodal distribution.

Problem

• zero-mean feature functions
• multimodal distributions
• Monte Carlo estimation
• control variates
• variance reduction

Innovation

• zero-mean feature functions
• control variates
• multimodal distributions
• density ratio
• variance reduction