Hardcastle, Luke; (2025) New approaches to survival extrapolation with inference using piecewise deterministic Monte Carlo. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

This thesis is primarily concerned with developing novel survival models that are able to extrapolate hazards beyond final event times, and the development of novel, efficient posterior sampling methods based on non-reversible processes. Polyhazard models are a class of flexible parametric models for modelling survival over extended time horizons. Significant user input is required, however, in selecting the number of latent hazards to model, their distributions and the choice of which variables to associate with each hazard. The resulting set of models is too large to explore manually, limiting their practical usefulness. To address this we extend the standard polyhazard model through a prior structure allowing for joint inference of parameters and structural quantities. The piecewise exponential model utilises a piecewise constant hazard function. We develop a novel extension to this model to allow for principled extrapolations, based on a two part prior: i) A discretisation of an underlying diffusion process, allowing prior information to inform extrapolations. ii) A Poisson point process prior for the set of knots, allowing this set to be extrapolated beyond final event times. Posterior inference in both cases is achieved using Markov Chain Monte Carlo methods based on Piecewise Deterministic Markov Processes. These processes have seen significant theoretical interest due to non-reversible dynamics allowing for efficient exploration of the state space, and tractable continuous trajectories that allow for efficient sampling from transdimensional posteriors. This thesis provides a literature review for the current state of these processes and makes several contributions to improving their implementation, including extending methods for generating the underlying Poisson process and extending the range of transdimensional posteriors they can be applied to. With respect to the latter, we develop theory that allows these processes to navigate posteriors comprised of mixtures of a manifold and the ambient space the manifold is embedded in.

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