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

Abstract

Abstract 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 $$\hbox {MSE}^{-1}$$ MSE - 1 , while single-level methods require $$\hbox {MSE}^{-\xi }$$ MSE - ξ for $$\xi >1$$ ξ > 1 . This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where $$\xi =5/4$$ ξ = 5 / 4 and $$\xi =3/2$$ ξ = 3 / 2 , respectively. It is also illustrated on more challenging log-Gaussian process models, where single-level complexity is approximately $$\xi =9/4$$ ξ = 9 / 4 and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives $$\xi = 5/4 + \omega $$ ξ = 5 / 4 + ω , for any $$\omega > 0$$ ω > 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.