Black box probabilistic numerics

Peer reviewe

Black box probabilistic numerics

Peer reviewe

Bayesian Quadrature with Prior Information: Modeling and Policies

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

Robotic Haptic Exploration of Shape and Symmetry

This thesis presents research on the use of symmetric models during haptic exploration procedures that have the objective of determining an object’s shape. These haptic exploration techniques, and their subsequent determination of a surface’s geometric properties, are crucial to allow robots to interact with a greater variety of objects, especially as the field of robotics transitions into unstructured environments. Symmetry is an extremely frequent shape property, especially in man-made objects, and it provides shape information that becomes useful in grasping and manipulation tasks, as well as enriching shape information for the aforementioned haptic exploration tasks. In this work, we present an improvement to Gaussian Process-driven exploration tasks. This method allows to describe symmetry to obtain a more precise shape estimation during active exploration, and can even be discovered in real time during the exploration procedure itself. This work involved the creation of a custom software resource to perform Gaussian Process regression with the addition of symmetries, and include a novel method of representing rotational symmetries. These novel models were then used in shape exploration procedures of 2D and 3D surfaces, both in a simulated environment and in an actual robotic task, using a series of custom-made contact sensors. These procedures are able to discover symmetry of each particular object in real time. This property can also be exploited, resulting in shape estimations that have a lower surface error and uncertainty. Additionally, exploration experiments that use these symmetry-finding procedures also require a lower total number of physical contacts and take less time to finish

Numerical Integration as and for Probabilistic Inference

Numerical integration or quadrature is one of the workhorses of modern scientific computing and a key operation to perform inference in intractable probabilistic models. The epistemic uncertainty about the true value of an analytically intractable integral identifies the integration task as an inference problem itself. Indeed, numerical integration can be cast as a probabilistic numerical method known as Bayesian quadrature (BQ). BQ leverages structural assumptions about the function to be integrated via properties encoded in the prior. A posterior belief over the unknown integral value emerges by conditioning the BQ model on an actively selected point set and corresponding function evaluations. Iterative updates to the Bayesian model turn BQ into an adaptive quadrature method that quantifies its uncertainty about the solution of the integral in a principled way. This thesis traces out the scope of probabilistic integration methods and highlights types of integration tasks that BQ excels at. These arise when sample efficiency is required and encodable prior knowledge about the integration problem of low to moderate dimensionality is at hand. The first contribution addresses transfer learning with BQ. It extends the notion of active learning schemes to cost-sensitive settings where cheap approximations to an expensive-to-evaluate integrand are available. The degeneracy of acquisition policies in simple BQ is lifted upon generalization to the multi-source, cost-sensitive setting. This motivates the formulation of a set of desirable properties for BQ acquisition functions. A second application considers integration tasks arising in statistical computations on Riemannian manifolds that have been learned from data. Unsupervised learning algorithms that respect the intrinsic geometry of the data rely on the repeated estimation of expensive and structured integrals. Our custom-made active BQ scheme outperforms conventional integration tools for Riemannian statistics. Despite their unarguable benefits, BQ schemes provide limited flexibility to construct suitable priors while keeping the inference step tractable. In a final contribution, we identify the ubiquitous integration problem of computing multivariate normal probabilities as a type of integration task that is structurally taxing for BQ. The instead proposed method is an elegant algorithm based on Markov chain Monte Carlo that permits both sampling from and estimating the normalization constant of linearly constrained Gaussians that contain an arbitrarily small probability mass. As a whole, this thesis contributes to the wider goal of advancing integration algorithms to satisfy the needs imposed by contemporary probabilistic machine learning applications