Moving Target Monte Carlo

TL;DR

The paper introduces Moving Target Monte Carlo (MTMC), a new sampling algorithm that approximates the posterior distribution iteratively, reducing costly evaluations by updating the acceptance rate based on an approximation that converges to the true posterior.

Contribution

MTMC is a novel non-Markovian sampling method that constructs acceptance rates using an iteratively refined approximation of the posterior, improving efficiency in intractable scenarios.

Findings

Proves convergence of the approximation to the true posterior.

Provides convergence rate estimates under various conditions.

Demonstrates reduced computational cost compared to traditional MCMC.

Abstract

The Markov Chain Monte Carlo (MCMC) methods are popular when considering sampling from a high-dimensional random variable $\mathbf{x}$ with possibly unnormalised probability density $p$ and observed data $\mathbf{d}$. However, MCMC requires evaluating the posterior distribution $p(\mathbf{x}|\mathbf{d})$ of the proposed candidate $\mathbf{x}$ at each iteration when constructing the acceptance rate. This is costly when such evaluations are intractable. In this paper, we introduce a new non-Markovian sampling algorithm called Moving Target Monte Carlo (MTMC). The acceptance rate at $n$-th iteration is constructed using an iteratively updated approximation of the posterior distribution $a_n(\mathbf{x})$ instead of $p(\mathbf{x}|\mathbf{d})$. The true value of the posterior $p(\mathbf{x}|\mathbf{d})$ is only calculated if the candidate $\mathbf{x}$ is accepted. The approximation $a_n$ utilises these evaluations and converges to $p$ as $n \rightarrow \infty$. A proof of convergence and estimation of convergence rate in different situations are given.