Bayesian Static Parameter Estimation for Partially Observed Diffusions using Multilevel Monte Carlo
19th October 2018, 3:30 pm – 4:30 pm
Main Maths Building, SM3
This talk will consider estimating parameters for partially observed SDE, which is known to be a challenging problem. A popular class of methods for solving this problem are particle MCMC (pMCMC) methods, such as particle marginal Metropolis-Hastings. Such methods leverage non-negative unbiased estimators of the marginal likelihood from a particle filter conditioned on a given parameter value within a pseudo-marginal MCMC algorithm in order to obtain an asymptotically exact algorithm without ever computing the marginal likelihood exactly. Here we assume furthermore that the SDE giving rise to the hidden process cannot be solved exactly, and must be approximated at finite resolution. It is well-known that in such contexts the multilevel Monte Carlo (MLMC) method can be used to substantially reduce the cost to achieve a given level of error. The idea is to represent the target expectation as a telescopic sum of increments of increasing cost, and estimate the increments using targets which are coupled in such a way that the increments have decreasing variance. A schedule of decreasing sample numbers can then be carefully constructed based upon the relationship between the variance and the cost, resulting in a substantial speedup. In the context of interest here it is not clear how to construct an exact coupling, and we instead appeal to a carefully constructed approximate coupling of the pairs of particle filters. It will be shown how to construct a consistent estimator with optimal speedup (i.e., the canonical Monte Carlo rate) via the approximate coupling.