Sequential Bayesian Design with Laplace Policies

Abstract

Since the formalisation of its foundational principles by Fisher (1935), experimental design has evolved extensively - recently through integration with fields such as machine learning (ML). This has led to the emergence of policy-based methods for sequential Bayesian design, where the policy, a parametric function, selects the next design point based on the current state of knowledge. The main contributions of this thesis are centred on the proposal of a novel Laplace-based policy parameterisation to represent current knowledge. The training of such Laplace-parameterised (LP) policies is supported through the development of a methodological framework, specifically optimised to facilitate efficient computation of mode derivatives. Also presented is an extension to the proposed LP-policy method, enabling its application to binary response models. This extension, necessitated by the non-differentiability of discrete simulations, utilises relaxed models that are partially implemented via concrete distributions, offering a novel approach to permit discrete responses within a policy gradient (PG) framework. With the Laplace parameterisation, the policy input comprises the mean and Hessian of a Laplace approximation to the posterior, which is updated as a sequential experiment progresses. This parameterisation of the policy provides an intuitive and compact knowledge representation, built upon established statistical principles. This contrasts with existing approaches that instead rely on simpler representations (e.g. raw trajectory data) or non-intuitive/black box policy parameterisations. The LP-policy is trained on an approximation to total expected information gain (EIG) using policy gradients, computed via automatic differentiation. A linear-Gaussian example verifies the proposed LP-policy approach as a viable method to sequential Bayesian design, which on average generates closer to optimal designs compared to alternative methods. The LP-policy approach also demonstrates superior performance when applied to a logistic regression example, yielding higher expected utility estimates than alternative approaches, including a state-of-the-art policy method that does not explicitly incorporate statistical principles into its policy parameterisation. Overall, the superior performance of the proposed LP-policy highlights the value of integrating established statistical principles within ML-driven approaches to experimental design. While other approaches favour end-to-end learning, this research demonstrates that explicitly leveraging statistical structure, rather than relying on it to be inferred by the policy, can lead to more effective and interpretable policies.

Date of Award	10 Nov 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Simon Cotter (Co Supervisor) & Timothy Waite (Main Supervisor)