Layered Adaptive Importance Sampling

TL;DR

This paper introduces a hierarchical adaptive importance sampling framework that automatically generates effective proposal densities, combining importance sampling and MCMC techniques to improve Monte Carlo estimations of complex integrals.

Contribution

It proposes a layered hierarchical procedure for adaptive importance sampling that guarantees suitable proposal densities and integrates multiple proposals with MCMC-driven adaptation.

Findings

Automatically generates effective proposal densities

Combines importance sampling with MCMC methods

Enhances Monte Carlo estimation accuracy

Abstract

Monte Carlo methods represent the "de facto" standard for approximating complicated integrals involving multidimensional target distributions. In order to generate random realizations from the target distribution, Monte Carlo techniques use simpler proposal probability densities to draw candidate samples. The performance of any such method is strictly related to the specification of the proposal distribution, such that unfortunate choices easily wreak havoc on the resulting estimators. In this work, we introduce a layered (i.e., hierarchical) procedure to generate samples employed within a Monte Carlo scheme. This approach ensures that an appropriate equivalent proposal density is always obtained automatically (thus eliminating the risk of a catastrophic performance), although at the expense of a moderate increase in the complexity. Furthermore, we provide a general unified importance sampling (IS) framework, where multiple proposal densities are employed and several IS schemes are introduced by applying the so-called deterministic mixture approach. Finally, given these schemes, we also propose a novel class of adaptive importance samplers using a population of proposals, where the adaptation is driven by independent parallel or interacting Markov Chain Monte Carlo (MCMC) chains. The resulting algorithms efficiently combine the benefits of both IS and MCMC methods.