Multi-index Sequential Monte Carlo ratio estimators for Bayesian Inverse problems

TL;DR

This paper introduces a multi-index Sequential Monte Carlo ratio estimator for Bayesian inverse problems, achieving optimal complexity and efficiency in estimating expectations with intractable models, outperforming traditional methods.

Contribution

The paper proposes a novel multi-index SMC method that combines MIMC and SMC advantages, providing provably optimal complexity for Bayesian inverse problems with PDE models.

Findings

Achieves canonical MSE$^{-1}$ complexity, outperforming single level methods.

Demonstrates efficiency on PDE-based Bayesian inverse problems in 1D and 2D.

Shows superiority over multilevel Monte Carlo in complex Gaussian process models.

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. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index Sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of Sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of MSE$^{-1}$, while single level methods require MSE$^{-\xi}$ for $\xi>1$. This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in $1$ and $2$ spatial dimensions, where $\xi=5/4$ and $\xi=3/2$, respectively. It is also illustrated on a more challenging log Gaussian process models, where single level complexity is approximately $\xi=9/4$ and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives $\xi = 5/4 + \omega$, for any $\omega > 0$, whereas our method is again canonical.