Title: 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. and 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) collapsing sum 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 method 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.