Bayesian inference and prediction for the inhomogeneous Poisson process, and a robust competitor to Hamiltonian Monte Carlo

Urbas, Szymon and Sherlock, Chris (2022) Bayesian inference and prediction for the inhomogeneous Poisson process, and a robust competitor to Hamiltonian Monte Carlo. PhD thesis, Lancaster University.

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Abstract

Modern Bayesian statistical methods utilise a plethora of mathematical and computational techniques to tackle real-world problems. The probabilistic formulation of a given model and its parameters can be used for making reliable predictions about future observations whilst accounting for parameter uncertainty. This, however, comes at a cost where the posterior distribution of the model parameters is not always tractable. To overcome intractability, computationally-intensive methods for Bayesian inference are employed; often, these need to be picked on a case-by-case basis, with algorithms themselves requiring careful tuning of the respective tuning parameters. This thesis explores three separate problems in Bayesian modelling and inference. In all three parts, the proposed solutions are based on computational methods relying on the simulation of particular stochastic processes. The first contribution tackles the problem of modelling and predicting recruitment to clinical trials. Phase III trials usually involve large recruitment drives across hundreds of centres to enrol a target number of patients in a prespecified period. It is crucial to accurately monitor the recruitment process at interim points of a study. Early detection of poor recruitment can inform trial organisers when making changes III to the protocol. We introduce a new flexible hierarchical model for multi-centre recruitment based on the inhomogeneous Poisson process. The simple, yet flexible, form of the model allows for efficient and user-friendly inference carried out in the Bayesian paradigm. The proposed framework outperforms state-of-the-art methods for recruitment prediction and is robust to model misspecifications. The second contribution is in the area of exact Cox process inference. A Cox process is a Poisson point process where the intensity itself is stochastic. Bayesian inference for Cox process models is an inherently difficult problem as the likelihood function of the intensity given observed data is not available in a closed form; this is known as a doubly-intractable problem. We introduce a novel unbiased estimator of the Cox process likelihood for processes with bounded intensity. The estimator can be implemented inside a random-weight particle filter to carry out exact inference (in a Monte Carlo sense) on the intensity function. In addition, it allows for efficient inference of the model hyperparameters through the pseudo-marginal Metropolis-Hastings algorithm. The third contribution is a competitor to the Hamiltonian Monte Carlo (HMC) algorithm. HMC is commonly used for problems in computational physics and Bayesian inference. It is a gradient-based algorithm, ideal for sampling from complex and high-dimensional targets. The main caveat, however, is the algorithm's sensitivity to one of the tuning parameters. In this thesis, we introduce a new algorithm, the Apogee to Apogee Path Sampler, which is based on HMC and thus benefits from many of its desirable properties. We show it has comparable efficiency to HMC but is much more robust to the misspecification of its tuning parameters.