Bayesian Static Parameter Estimation for Partially Observed Diffusions via Multilevel Monte Carlo

Abstract

In this article we consider static Bayesian parameter estimation for partially observed diffusions that are discretely observed. We work under the assumption that one must resort to discretizing the underlying diffusion process, for instance using the Euler Maruyama method.Given this assumption, we show how one can use Markov chain Monte Carlo (MCMC) and particularly particle MCMC [Andrieu, C., Doucet, A. & Holenstein, R. (2010). Particle Markov chain Monte Carlo methods (with discussion). J. R. Statist. Soc. Ser. B, 72, 269-342] to implement a new approximation of the multilevel (ML) Monte Carlo (MC) collapsingsum identity. Our approach comprises constructing an approximate coupling of the posterior density of the joint distribution over parameter and hidden variables at two different discretization levels and then correcting by an importance sampling method. The variance of the weights are independent of the length of the observed data set. The utility of such a method is that, for a prescribed level of mean square error, the cost of this MLMC is provably less than i.i.d. sampling from the posterior associated to the most precise discretization. However the method here comprises using only known and efficient simulation methodologies. The theoretical results are illustrated by inference of the parameters of two prototypical processes given noisy partial observations of the process: the first is an Ornstein Uhlenbeck process and the second is a more general Langevin equation.

A Multi-Index Markov Chain Monte Carlo Method

“…It is noted that in the case of a single discretized dimension the method presented constitutes a new Multilevel Markov chain Monte Carlo (MLMCMC) method, which generalizes [11]. Furthermore, in this case the proof of Proposition 3.1 goes through for the general estimator (10) rather than the simplified one with known normalization constants (11). There exist 2 other general MLMCMC methods in the literature.…”

“…The requirement to of independent (or exact) sampling from couples with the correct marginals is often not possible in many contexts. This has been dealt with in several recent works, such as [2,11,12,13].…”

Multilevel Markov Chain Monte Carlo

“…The motivation is typically the application to Bayesian inference problems or to sequential inference problems arising in the context of data assimilation and filtering. This includes the Metropolis-Hastings type Markov chain Monte Carlo estimators considered in this work, as well as importance sampling based multilevel Markov chain Monte Carlo methods [42], multilevel sequential Monte Carlo methods and multilevel particle filters [35,3,45,20,49,46], multilevel ensemble Kalman filters [44], multilevel approximate Bayesian computation [61], multilevel stochastic gradient Markov chain Monte Carlo algorithms [32] and multilevel methods based on ratio estimators [54,21]. We would also like to point out here that the "levels" in the references above are not necessarily defined through a hierarchy of meshes, but can also involve different numbers of particles (e.g.…”

A practical guide to pseudo-marginal methods for computational inference in systems biology

“…It is also important to note that the pseudo-marginal approach is equally valid for Bayesian sampling strategies based on sequential Monte Carlo (Del Moral et al, 2006;Li et al, 2019;Sisson et al, 2007). Furthermore, advances in stochastic simulation (Schnoerr et al, 2017;Warne et al, 2019) can also improve the performance of the likelihood estimator, and the application of multilevel Monte Carlo to particle filters can further reduce estimator variance (Gregory et al, 2016;Jasra et al, 2017Jasra et al, , 2018.…”