Bayesian Experimental Design Based on Mean Objective Cost of Uncertainty

Abstract

We propose several efficient algorithms for Bayesian experimental design when studying complex systems under uncertainty with specific operational objectives. Throughout this dissertation, the uncertainty is quantified by the mean objective cost of uncertainty (MOCU).

First, we develop MOCU-based experimental design for physical systems described by Stochastic Differential Equations (SDEs) with uncertain model parameters. We assume the observed signals are from a system whose dynamics is governed by SDEs. The observations can be degraded by blurring and additive noise. We aim to derive a optimal robust filter minimizing the expected filtering error. We further derive an optimal experimental design framework to determine the importance of the SDE parameters. Such a framework can update the knowledge about the system and thereafter the signal processes. As a result, it guides the systems knowledge discovery to help derive better filters.

In the MOCU-based framework, we further study Bayesian active learning to sequentially sample queries to improve predictive models. For classification, the goal is to learn the optimal classifier with high prediction accuracy when classification labels is difficult or costly to obtain. The MOCU-based active learning procedure is shown to get stuck before converging to the optimal classifier, due to the piece-wise linearity of MOCU. We propose two methods to address this myopic issue of MOCU-based active learning by approximating the MOCU functions with strict concavities, named Weighted-MOCU (WMOCU) and Soft-MOCU (SMOCU). We provide theoretical proofs of the convergence of these methods and demonstrate their sampling efficiency with both synthetic and real-world experiments.

Finally, to explore more practical MOCU-based experimental design, we study MOCU-based active learning of both pool-based and query synthetic scenarios for Gaussian Process Classification (GPC). We develop computationally efficient algorithms for MOCU-based active learning with GPC. Our algorithms compute the joint predictive distribution of label pairs as a one-dimensional integral, which enables us to compute MOCU-based acquisition functions without incrementally retraining GPC for each possible query. By deriving the gradient chain rule to efficiently calculate the gradient of SMOCU reduction, we also develop the first MOCU-based query synthesis active learning algorithm.