MULTILEVEL SMC2 FOR BAYESIAN INFERENCE OF PARTIALLY OBSERVED DIFFUSIONS

Abstract

In this article, we consider the Bayesian inference in the hidden Markov Model given by partially observed diffusions. For Euler discretized partially observed diffusions, SMC2 designed by Chopin et al. (2012) would be an efficient tool for sequential analysis. Our approach combines SMC2 and the Multilevel Monte Carlo (MLMC) methods introduced by Jasra et al. (2017) to implement a new algorithm for Bayesian inference. The idea is to approximate each summand of the MLMC summation by simulating from a dependent coupling of the posterior density at two different discretization levels and rectifying with importance sampling methods. Compared to the single level SMC2 which i.i.d. samples from the posterior density associated to a chosen discretization level, this multilevel SMC2 algorithm could have less cost than the single level algorithm with the most precise discretization for a given level of mean square error. The theoretical results are supported by the numerical simulations of Langevin equation.