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

## TL;DR

This paper introduces a multilevel Monte Carlo approach combined with particle MCMC for efficient Bayesian parameter estimation in discretized partially observed diffusions, reducing computational cost while maintaining accuracy.

## Contribution

It develops a novel MLMC method using coupling and importance sampling for Bayesian inference in discretized diffusions, improving efficiency over traditional methods.

## Key findings

- Variance of weights is independent of data length
- Method reduces computational cost for a given mean square error
- Successfully applied to Ornstein-Uhlenbeck and Langevin processes

## 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.