Accelerating pseudo-marginal metropolis-hastings schemes for stochastic kinetic models

Abstract

Stochastic kinetic models (SKMs) provide a natural framework for modelling continuoustime physical processes with inherent stochasticity. As such, they are frequently used to model interacting species populations in areas such as epidemiology, population ecology and systems biology. This thesis focuses on the challenging problem of performing fully Bayesian inference for the rate constants governing these models, using discrete-time observations of the species populations that may be incomplete and subject to measurement error. The SKM is often represented by either a Markov jump process (MJP) or an Itˆo diffusion process. In either case, the observed data likelihood is intractable, necessitating the use of computationally intensive techniques such as pseudo-marginal Metropolis-Hastings (PMMH). One prominent example of PMMH is particle Markov chain Monte Carlo (particle MCMC), whereby the observed data likelihood is unbiasedly estimated using a particle filter. Whilst powerful, such schemes are often impractical due to their large computational expense. This thesis aims to increase the computational and statistical efficiency of these schemes using various techniques. Several of these techniques leverage a tractable surrogate model, the linear noise approximation (LNA), which can be derived directly from the MJP or the diffusion process. The LNA can be used in three ways: in the design of a gradient-based parameter proposal such as the Metropolis-adjusted Langevin algorithm (MALA); in the first stage of a delayed-acceptance step; and to construct an appropriate bridge construct within the particle filter. Further computational savings can be made if several of these techniques are used in tandem, as the equations governing the LNA need only be solved once for use in all three techniques. A further acceleration technique involves inducing positive correlation between successive likelihood estimates within the particle filter. A novel approach to MALA is also proposed, whereby the gradient is approximated to reduce the number of differential equations required to estimate it. The proposed acceleration techniques are then applied to several models utilising real-world and synthetic data, to compare their performance.