A randomized multi-index sequential Monte Carlo method

TL;DR

This paper introduces a randomized multi-index Sequential Monte Carlo method that eliminates bias and maintains optimal complexity for estimating expectations, even when the target distribution is approximated at finite resolution.

Contribution

It extends multi-index SMC with a randomization strategy to remove bias, simplifying implementation and preserving efficiency in complex inference tasks.

Findings

Achieves the same MSE$^{-1}$ complexity as the original method

Removes discretization bias through randomization

Demonstrates effectiveness on Bayesian inverse and spatial statistics problems

Abstract

We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. Under such an assumption, this work builds upon a recently introduced multi-index Sequential Monte Carlo (SMC) ratio estimator, which provably enjoys the complexity improvements of multi-index Monte Carlo (MIMC) and the efficiency of SMC for inference. The present work leverages a randomization strategy to remove bias entirely, which simplifies estimation substantially, particularly in the MIMC context, where the choice of index set is otherwise important. Under reasonable assumptions, the proposed method provably achieves the same canonical complexity of MSE$^{-1}$ as the original method (where MSE is mean squared error), but without discretization bias. It is illustrated on examples of Bayesian inverse and spatial statistics problems.