Date of Award
Doctor of Philosophy (PhD)
Quadrature is the problem of estimating intractable integrals. Such integrals regularly arise in engineering and the natural sciences, especially when Bayesian methods are applied; examples include model evidences, normalizing constants and marginal distributions. This dissertation explores Bayesian quadrature, a probabilistic, model-based quadrature method. Specifically, we study different ways in which Bayesian quadrature can be adapted to account for different kinds of prior information one may have about the task. We demonstrate that by taking into account prior knowledge, Bayesian quadrature can outperform commonly used numerical methods that are agnostic to prior knowledge, such as Monte Carlo based integration. We focus on two types of information that are (a) frequently available when faced with an intractable integral and (b) can be (approximately) incorporated into Bayesian quadrature:
• Natural bounds on the possible values that the integrand can take, e.g., when the integrand is a probability density function, it must nonnegative everywhere.• Knowledge about how the integral estimate will be used, i.e., for settings where quadrature is a subroutine, different downstream inference tasks can result in different priorities or desiderata for the estimate.
These types of prior information are used to inform two aspects of the Bayesian quadrature inference routine:
• Modeling: how the belief on the integrand can be tailored to account for the additional information.• Policies: where the integrand will be observed given a constrained budget of observations.
This second aspect of Bayesian quadrature, policies for deciding where to observe the integrand, can be framed as an experimental design problem, where an agent must choose locations to evaluate a function of interest so as to maximize some notion of value. We will study the broader area of sequential experimental design, applying ideas from Bayesian decision theory to develop an efficient and nonmyopic policy for general sequential experimental design problems. We consider other sequential experimental design tasks such as Bayesian optimization and active search; in the latter, we focus on facilitating human–computer partnerships with the goal of aiding human agents engaged in data foraging through the use of active search based suggestions and an interactive visual interface. Finally, this dissertation will return to Bayesian quadrature and discuss the batch setting for experimental design, where multiple observations of the function in question are made simultaneously.
Sanmay Das, Chien-Ju Ho, Chris Oates, Netanel Raviv,