# Stratification as a general variance reduction method for Markov chain Monte Carlo

## TL;DR

This paper presents a theoretical analysis of EMUS, a stratified sampling method, demonstrating its efficiency in computing averages over complex distributions, especially in multimodal or tail probability scenarios.

## Contribution

The paper provides a detailed theoretical foundation for EMUS, establishing it as a versatile and more efficient alternative to traditional MCMC in challenging sampling problems.

## Key findings

- EMUS can outperform MCMC in multimodal distributions.
- EMUS is effective for tail probability estimation.
- Theoretical analysis confirms EMUS's efficiency.

## Abstract

The Eigenvector Method for Umbrella Sampling (EMUS) belongs to a popular class of methods in statistical mechanics which adapt the principle of stratified survey sampling to the computation of free energies. We develop a detailed theoretical analysis of EMUS. Based on this analysis, we show that EMUS is an efficient general method for computing averages over arbitrary target distributions. In particular, we show that EMUS can be dramatically more efficient than direct MCMC when the target distribution is multimodal or when the goal is to compute tail probabilities. To illustrate these theoretical results, we present a tutorial application of the method to a problem from Bayesian statistics.