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

Abstract We consider the problem of estimating expectations with respect to a target distribution an unknown normalising constant, and where even un-normalised needs be approximated at finite resolution. This setting is ubiquitous across science engineering applications, for example in context Bayesian inference physics-based model governed by intractable partial differential equation (PDE) appears likelihood. A multi-index sequential Monte Carlo (MISMC) method used construct ratio estimators which provably enjoy complexity improvements (MIMC) as well efficiency (SMC) inference. In particular, proposed achieves canonical $$\hbox {MSE}^{-1}$$ MSE - 1 , while single-level methods require {MSE}^{-\xi }$$ ξ $$\xi >1$$ > . illustrated on examples inverse problems elliptic PDE forward 1 2 spatial dimensions, =5/4$$ = 5 / 4 =3/2$$ 3 2 respectively. It also more challenging log-Gaussian process models, approximately =9/4$$ 9 multilevel (or MIMC inappropriate index set) gives = 5/4 + \omega $$ + ω any $$\omega > 0$$ 0 whereas our again canonical. provide novel theoretical verification product-form convergence results requires Gaussian processes built spaces mixed regularity defined spectral domain, facilitates acceleration fast Fourier transform via cumulant embedding strategy, may independent interest statistics machine learning.