Augmented pseudo-marginal Metropolis-Hastings for partially observed diffusion processes

TL;DR

This paper introduces an augmented pseudo-marginal Metropolis-Hastings algorithm for efficient inference in complex diffusion processes with incomplete and noisy data, avoiding resampling issues and improving computational efficiency.

Contribution

The authors propose a novel augmentation scheme that conditions on latent process values, eliminating resampling and enhancing the efficiency of inference algorithms for diffusion SDEs.

Findings

Significant efficiency improvements over existing methods.

Effective handling of incomplete and noisy data.

Unified framework combining Gibbs sampling and pseudo-marginal schemes.

Abstract

We consider the problem of inference for nonlinear, multivariate diffusion processes, satisfying It\^o stochastic differential equations (SDEs), using data at discrete times that may be incomplete and subject to measurement error. Our starting point is a state-of-the-art correlated pseudo-marginal Metropolis-Hastings algorithm, that uses correlated particle filters to induce strong and positive correlation between successive likelihood estimates. However, unless the measurement error or the dimension of the SDE is small, correlation can be eroded by the resampling steps in the particle filter. We therefore propose a novel augmentation scheme, that allows for conditioning on values of the latent process at the observation times, completely avoiding the need for resampling steps. We integrate over the uncertainty at the observation times with an additional Gibbs step. Connections between the resulting pseudo-marginal scheme and existing inference schemes for diffusion processes are made, giving a unified inference framework that encompasses Gibbs sampling and pseudo marginal schemes. The methodology is applied in three examples of increasing complexity. We find that our approach offers substantial increases in overall efficiency, compared to competing methods.