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

Automating Active Learning for Gaussian Processes

In many problems in science, technology, and engineering, unlabeled data is abundant but acquiring labeled observations is expensive -- it requires a human annotator, a costly laboratory experiment, or a time-consuming computer simulation. Active learning is a machine learning paradigm designed to minimize the cost of obtaining labeled data by carefully selecting which new data should be gathered next. However, excessive machine learning expertise is often required to effectively apply these techniques in their current form. In this dissertation, we propose solutions that further automate active learning. Our core contributions are active learning algorithms that are easy for non-experts to use but that deliver results competitive with or better than human-expert solutions. We begin introducing a novel active search algorithm that automatically and dynamically balances exploration against exploitation --- without relying on a parameter to control this tradeoff. We also provide a theoretical investigation on the hardness of this problem, proving that no polynomial-time policy can achieve a constant factor approximation ratio for the expected utility of the optimal policy. Next, we introduce a novel information-theoretic approach for active model selection. Our method is based on maximizing the mutual information between the output variable and the model class. This is the first active-model-selection approach that does not require updating each model for every candidate point. As a result, we successfully developed an automated audiometry test for rapid screening of noise-induced hearing loss, a widespread and preventable disability, if diagnosed early. We proceed by introducing a novel model selection algorithm for fixed-size datasets, called Bayesian optimization for model selection (BOMS). Our proposed model search method is based on Bayesian optimization in model space, where we reason about the model evidence as a function to be maximized. BOMS is capable of finding a model that explains the dataset well without any human assistance. Finally, we extend BOMS to active learning, creating a fully automatic active learning framework. We apply this framework to Bayesian optimization, creating a sample-efficient automated system for black-box optimization. Crucially, we account for the uncertainty in the choice of model; our method uses multiple and carefully-selected models to represent its current belief about the latent objective function. Our algorithms are completely general and can be extended to any class of probabilistic models. In this dissertation, however, we mainly use the powerful class of Gaussian process models to perform inference. Extensive experimental evidence is provided to demonstrate that all proposed algorithms outperform previously developed solutions to these problems

Domain-Agnostic Batch Bayesian Optimization with Diverse Constraints via Bayesian Quadrature

Real-world optimisation problems often feature complex combinations of (1) diverse constraints, (2) discrete and mixed spaces, and are (3) highly parallelisable. (4) There are also cases where the objective function cannot be queried if unknown constraints are not satisfied, e.g. in drug discovery, safety on animal experiments (unknown constraints) must be established before human clinical trials (querying objective function) may proceed. However, most existing works target each of the above three problems in isolation and do not consider (4) unknown constraints with query rejection. For problems with diverse constraints and/or unconventional input spaces, it is difficult to apply these techniques as they are often mutually incompatible. We propose cSOBER, a domain-agnostic prudent parallel active sampler for Bayesian optimisation, based on SOBER of Adachi et al. (2023). We consider infeasibility under unknown constraints as a type of integration error that we can estimate. We propose a theoretically-driven approach that propagates such error as a tolerance in the quadrature precision that automatically balances exploitation and exploration with the expected rejection rate. Moreover, our method flexibly accommodates diverse constraints and/or discrete and mixed spaces via adaptive tolerance, including conventional zero-risk cases. We show that cSOBER outperforms competitive baselines on diverse real-world blackbox-constrained problems, including safety-constrained drug discovery, and human-relationship-aware team optimisation over graph-structured space.Comment: 24 pages, 5 figure

Multi-Period Asset Allocation: An Application of Discrete Stochastic Programming

The issue of modeling farm financial decisions in a dynamic framework is addressed in this paper. Discrete stochastic programming is used to model the farm portfolio over the planning period. One of the main issues of discrete stochastic programming is representing the uncertainty of the data. The development of financial scenario generation routines provides a method to model the stochastic nature of the model. In this paper, two approaches are presented for generating scenarios for a farm portfolio problem. The approaches are based on copulas and optimization. The copula method provides an alternative to the multivariate normal assumption. The optimization method generates a number of discrete outcomes which satisfy specified statistical properties by solving a non-linear optimization model. The application of these different scenario generation methods is then applied to the topic of geographical diversification. The scenarios model the stochastic nature of crop returns and land prices in three separate geographic regions. The results indicate that the optimal diversification strategy is sensitive to both scenario generation method and initial acreage assumptions. The optimal diversification results are presented using both scenario generation methods.Agribusiness, Agricultural Finance, Farm Management,

PriorCVAE: scalable MCMC parameter inference with Bayesian deep generative modelling

In applied fields where the speed of inference and model flexibility are crucial, the use of Bayesian inference for models with a stochastic process as their prior, e.g. Gaussian processes (GPs) is ubiquitous. Recent literature has demonstrated that the computational bottleneck caused by GP priors or their finite realizations can be encoded using deep generative models such as variational autoencoders (VAEs), and the learned generators can then be used instead of the original priors during Markov chain Monte Carlo (MCMC) inference in a drop-in manner. While this approach enables fast and highly efficient inference, it loses information about the stochastic process hyperparameters, and, as a consequence, makes inference over hyperparameters impossible and the learned priors indistinct. We propose to resolve the aforementioned issue and disentangle the learned priors by conditioning the VAE on stochastic process hyperparameters. This way, the hyperparameters are encoded alongside GP realisations and can be explicitly estimated at the inference stage. We believe that the new method, termed PriorCVAE, will be a useful tool among approximate inference approaches and has the potential to have a large impact on spatial and spatiotemporal inference in crucial real-life applications. Code showcasing the PriorCVAE technique can be accessed via the following link: https://github.com/elizavetasemenova/PriorCVA