Markov Kernels Local Aggregation for Noise Vanishing Distribution Sampling

TL;DR

This paper introduces a state-dependent kernel aggregation method for Markov Chain Monte Carlo that leverages local topology to improve convergence and reduce variance in noise vanishing distributions.

Contribution

It proposes a novel local aggregation strategy for Markov kernels that adapts to the target distribution's local structure, enhancing sampling efficiency.

Findings

Locally-weighted aggregation converges faster than random-scan methods.

The approach yields smaller asymptotic variances.

Theoretical and empirical evidence supports improved performance.

Abstract

A novel strategy that combines a given collection of $\pi$-reversible Markov kernels is proposed. At each Markov transition, one of the available kernels is selected via a state-dependent probability distribution. In contrast to random-scan type approaches that assume a constant (i.e. state-independent) selection probability distribution, the state-dependent distribution is specified so as to privilege moving according to a kernel which is relevant for the local topology of the target distribution. This approach leverages paths or other low dimensional manifolds that are typically present in noise vanishing distributions. Some examples for which we show (theoretically or empirically) that a locally-weighted aggregation converges substantially faster and yields smaller asymptotic variances than an equivalent random-scan algorithm are provided.