Multi-index Sequential Monte Carlo Ratio Estimators for Bayesian Inverse problems

We consider the problem of estimating expectations with respect to a target distribution with an unknown normalising constant, and where even the un-normalised 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 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. We also provide novel theoretical verification of the product-form convergence results which MIMC requires for Gaussian processes built in spaces of mixed regularity defined in the spectral domain, which facilitates acceleration with fast Fourier transform methods via a cumulant embedding strategy, and may be of independent interest in the context of spatial statistics and machine learning.

Automated summary

This paper discusses the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant. A multi-index sequential Monte Carlo method is proposed to improve the efficiency of inference, and it is illustrated on various examples to verify its effectiveness.