Unbiased inference for discretely observed hidden Markov model diffusions

TL;DR

This paper introduces a Bayesian inference method for discretely observed diffusion processes that eliminates discretisation bias without requiring exact simulation, using standard approximations and advanced Monte Carlo techniques.

Contribution

It presents a novel unbiased inference approach combining particle MCMC, multilevel Monte Carlo, and importance sampling, applicable to noisy discretely observed diffusions.

Findings

Method achieves bias-free inference as Markov chain iterations increase.

Approach is computationally efficient and easily parallelisable.

Demonstrated on Ornstein-Uhlenbeck, geometric Brownian motion, and Langevin dynamics.

Abstract

We develop a Bayesian inference method for diffusions observed discretely and with noise, which is free of discretisation bias. Unlike existing unbiased inference methods, our method does not rely on exact simulation techniques. Instead, our method uses standard time-discretised approximations of diffusions, such as the Euler--Maruyama scheme. Our approach is based on particle marginal Metropolis--Hastings, a particle filter, randomised multilevel Monte Carlo, and importance sampling type correction of approximate Markov chain Monte Carlo. The resulting estimator leads to inference without a bias from the time-discretisation as the number of Markov chain iterations increases. We give convergence results and recommend allocations for algorithm inputs. Our method admits a straightforward parallelisation, and can be computationally efficient. The user-friendly approach is illustrated on three examples, where the underlying diffusion is an Ornstein--Uhlenbeck process, a geometric Brownian motion, and a 2d non-reversible Langevin equation.