Bayesian optimal design using stochastic gradient optimisation and surrogate utility functions

Abstract

Experimental design is becoming increasingly important to many applications from genetic research to robotics. It provides a structured way of allocating resources in an e cient manner prior to an experiment being conducted. Assuming a model for the data, one approach is to introduce a utility function to quantify the worth of a design given some data and parameters. Typically experiment-speci c utility functions are di cult to elicit and hence a pragmatic choice of utility concerning the information gained about the model parameters is used. Bayesian experimental design aims to maximise the expected utility accounting for uncertainty in the model parameters and the data which could be observed. For this approach, di culties arise as the expected utility is typically intractable and computationally costly to approximate. Modern applications often seek high dimensional designs. In these settings existing algorithms such as the MCMC scheme of Müller (1999) and ACE (Overstall and Woods, 2017), require a high number of utility evaluations before they converge. For the most commonly used utility functions this becomes a computationally costly exercise. Therefore there is a need for an e cient and scalable method for nding the Bayesian optimal design. The contributions of this thesis are as follows. Firstly, stochastic gradient optimisation, a scalable method widely used in the eld of machine learning, is applied to the Bayesian experimental design problem. The second contribution is to consider a utility function based on the Fisher information matrix as a Bayesian utility function by showing it has a decision theoretic justi cation. These utilities are often available in a closed-form so are fast to compute. The nal contribution is to investigate surrogate functions for expensive utilities as an e cient way of nding promising regions of the design space.