Bayesian inference for continuous time Markov chains

Continuous time Markov chains (CTMCs) are a flexible class of stochastic models that have been employed in a wide range of applications from timing of computer protocols, through analysis of reliability in engineering, to models of biochemical networks in molecular biology. These models are defined as a state system with continuous time transitions between the states. Extensive work has been historically performed to enable convenient and flexible definition, simulation, and analysis of continuous time Markov chains. This thesis considers the problem of Bayesian parameter inference on these models and investigates computational methodologies to enable such inference. Bayesian inference over continuous time Markov chains is particularly challenging as the likelihood cannot be evaluated in a closed form. To overcome the statistical problems associated with evaluation of the likelihood, advanced algorithms based on Monte Carlo have been used to enable Bayesian inference without explicit evaluation of the likelihoods. An additional class of approximation methods has been suggested to handle such inference problems, known as approximate Bayesian computation. Novel Markov chain Monte Carlo (MCMC) approaches were recently proposed to allow exact inference. The contribution of this thesis is in discussion of the techniques and challenges in implementing these inference methods and performing an extensive comparison of these approaches on two case studies in systems biology. We investigate how the algorithms can be designed and tuned to work on CTMC models, and to achieve an accurate estimate of the posteriors with reasonable computational cost. Through this comparison, we investigate how to avoid some practical issues with accuracy and computational cost, for example by selecting an optimal proposal distribution and introducing a resampling step within the sequential Monte-Carlo method. Within the implementation of the ABC methods we investigate using an adaptive tolerance schedule to maximise the efficiency of the algorithm and in order to reduce the computational cost