Bayesian inference for stochastic processes

Malory, Sean and Sherlock, Chris (2021) Bayesian inference for stochastic processes. PhD thesis, Lancaster University.

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Abstract

This thesis builds upon two strands of recent research related to conducting Bayesian inference for stochastic processes. Firstly, this thesis will introduce a new residual-bridge proposal for approximately simulating conditioned diffusions formed by applying the modified diffusion bridge approximation of Durham and Gallant, 2002 to the difference between the true diffusion and a second, approximate, diffusion driven by the same Brownian motion. This new proposal attempts to account for volatilities which are not constant and can, therefore, lead to gains in efficiency over recently proposed residual-bridge constructs (Whitaker et al., 2017) in situations where the volatility varies considerably, as is often the case for larger interobservation times and for time-inhomogeneous volatilities. These gains in efficiency are illustrated via a simulation study for three diffusions; the Birth-Death (BD) diffusion, the Lotka-Volterra (LV) diffusion, and a diffusion corresponding to a simple model of gene expression (GE). Secondly, this thesis will introduce two new classes of Markov Chain Monte Carlo samplers, named the Exchangeable Sampler and the Exchangeable Particle Gibbs Sampler, which, at each iteration, use exchangeablility to simulate multiple, weighted proposals whose weights indicate how likely the chain is to move to such a proposal. By generalising the Independence Sampler and the Particle Gibbs Sampler respectively, these new samplers allow for the locality of moves to be controlled by a scaling parameter which can be tuned to optimise the mixing of the resulting MCMC procedure, while still benefiting from the increase in acceptance probability that typically comes with using multiple proposals. These samplers can lead to chains with better mixing properties, and, therefore, to MCMC estimators with smaller variances than their corresponding algorithms based on independent proposals. This improvement in mixing is illustrated, numerically, for both samplers through simulation studies, and, theoretically, for the Exchangeable Sampler through a result which states that, under certain conditions, the Exchangeable Sampler is geometrically ergodic even when the importance weights are unbounded and, hence, in scenarios where the Independence Sampler cannot be geometrically ergodic. To provide guidance in the practical implementation of such samplers, this thesis derives asymptotic expected squared-jump distance results for the Exchangeable Sampler and the Exchangeable Particle Gibbs Sampler. Moreover, simulation studies demonstrate, numerically, how the theory plays out in practice when d is finite.