Hierarchical experiments and likelihood approximations

Abstract

The objective of this thesis is the development of new methods for exploiting multi- level approximations in statistical modelling and design using hierarchical Gaussian processes.

An important application is in multi-level approximations of intractable likelihood problems, where a hierarchy of likelihood approximations is available, each having different accuracy and cost. For example, in generalised linear mixed models, quadrature methods may be used to approximate the likelihood, and by varying the number of quadrature points used we may trade off accuracy against computational cost. Multi-level Gaussian processes methods have been established and implemented to emulate the highest level of accuracy available, using data from all stages of the hierarchy. Moreover, Gaussian process emulated likelihoods are used to conduct inference about the model parameters and to address uncertainty.

There exists a general design problem for hierarchical Gaussian process: to decide how many times to evaluate each level of approximation, and at which parameter values, in order to find an accurate approximation to the point maximising that function, given a limited computational budget. A decision-theoretic approach, called expected gain in utility, based on Bayesian optimisation has been developed and demonstrated through examples.

This thesis is focused on likelihood approximations and the aim is to compute an accurate approximation to the likelihood and to the maximum likelihood estimate (the maximiser of the high-level likelihood approximation) at minimal cost. The methodology is demonstrated on generalised linear mixed model and Ising model examples throughout the thesis. While this thesis in mainly focused on likelihood approximation, the expected gain in utility methodology for choosing the experimental design could also be used on other applications of multi-level Gaussian processes.

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Identifiers

ORCID for Helen Ogden: orcid.org/0000-0001-7204-9776

ORCID for Dave Woods: orcid.org/0000-0001-7648-429X

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