Theoretical properties of quasi-stationary Monte Carlo methods

TL;DR

This paper establishes foundational theoretical results for quasi-stationary Monte Carlo methods, including conditions for convergence to target distributions and rates of convergence, with detailed analysis of a killed Ornstein-Uhlenbeck process.

Contribution

It provides the first rigorous conditions under which quasi-stationary distributions match target densities in Monte Carlo inference, and quantifies convergence rates.

Findings

Sufficient conditions for quasi-limiting distribution to match target density

Quantification of convergence rates via Langevin diffusion comparison

Detailed analysis of a killed Ornstein-Uhlenbeck process with Gaussian quasi-stationary distribution

Abstract

This paper gives foundational results for the application of quasi-stationarity to Monte Carlo inference problems. We prove natural sufficient conditions for the quasi-limiting distribution of a killed diffusion to coincide with a target density of interest. We also quantify the rate of convergence to quasi-stationarity by relating the killed diffusion to an appropriate Langevin diffusion. As an example, we consider in detail a killed Ornstein--Uhlenbeck process with Gaussian quasi-stationary distribution.