Bayesian inference for linear stochastic differential equations with application to biological processes

Abstract

Stochastic differential equations (SDEs) provide a natural framework for describing the stochasticity inherent in physical processes that evolve continuously over time. In this thesis, we consider the problem of Bayesian inference for a specific class of SDE – one in which the drift and diffusion coefficients are linear functions of the state. Although a linear SDE admits an analytical solution, the inference problem remains challenging, due to the absence of a closed form expression for the posterior density of the parameter of interest and any unobserved components. This necessitates the use of sampling-based approaches such as Markov chain Monte Carlo (MCMC) and, in cases where observed data likelihood is intractable, particle MCMC (pMCMC). When data are available on multiple experimental units, a stochastic differential equation mixed effects model (SDEMEM) can be used to further account for between-unit variation. Integrating over this additional uncertainty is computationally demanding. Motivated by two challenging biological applications arising from physiology studies of mice, the aim of this thesis is the development of efficient sampling-based inference schemes for linear SDEs. A key contribution is the development of a novel Bayesian inference scheme for SDEMEMs.