Unbiased Inference for Discretely Observed Hidden Markov Model Diffusions

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.

Unbiased Estimation of the Hessian for Partially Observed Diffusions

“…Specifically it will be a particle marginal Metropolis Hastings algorithm. We omit such a description of the later, as we only use it as a comparison, but we refer the reader to [11] for a more concrete description. However we emphasis with it, that it is only asymptotically unbiased, in relation to the Hessian identity (2.4).…”

“…This verifies the result in Lemma A.2. At the top of Figure (11), we present the log-log plot of cost against ∆ L for (3.9) and the R&G score estimate both with M = 10. The experiments is done over L ∈ {2, 3, 4, 5, 6, 7}.…”

“…In particular these methods allow to unbiasedly estimate an expectation of a functional, by randomizing on the level of the time-discretization in a type of multilevel Monte Carlo (MLMC) approach [32], where there is a coupling between different levels. As a result, this methodology has been considered in the context of both filtering and Bayesian computation [11,19,21] and gradient estimation [18].…”

“…We then describe how one can attain unbiased estimators, which is based on various couplings of the CPF. From this we test this methodology to that of using the methods of [11,22] for the Hessian computation, as for a comparison, where we demonstrate the unbiased estimator through both the variance and bias. This will be conducted on both a single and multidimensional Ornstein-Uhlenbeck process, as well as a more complicated model of the Fitzhugh-Nagumo model.…”

Unbiased estimation of the Hessian for partially observed diffusions

“…We then describe how one can attain unbiased estimators, which is based on various couplings of the CPF. From this, we test this methodology to that of using the methods of [18,29] for the Hessian computation, as for a comparison, where we demonstrate the unbiased estimator through both the variance and bias. This will be conducted on both a single and multidimensional Ornstein–Uhlenbeck (OU) process, as well as a more complicated form of the FHN model.…”