Bayesian Decision-theoretic Methods for Parameter Ensembles with Application to Epidemiology

Parameter ensembles or sets of random effects constitute one of the cornerstones of modern statistical practice. This is especially the case in Bayesian hierarchical models, where several decision theoretic frameworks can be deployed to optimise the estimation of parameter ensembles. The reporting of such ensembles in the form of sets of point estimates is an important concern in epidemiology, and most particularly in spatial epidemiology, where each element in these ensembles represent an epidemiological unit such as a hospital or a geographical area of interest. The estimation of these parameter ensembles may substantially vary depending on which inferential goals are prioritised by the modeller. Since one may wish to satisfy a range of desiderata, it is therefore of interest to investigate whether some sets of point estimates can simultaneously meet several inferential objectives. In this thesis, we will be especially concerned with identifying ensembles of point estimates that produce good approximations of (i) the true empirical quantiles and empirical quartile ratio (QR) and (ii) provide an accurate classification of the ensemble’s elements above and below a given threshold. For this purpose, we review various decision-theoretic frameworks, which have been proposed in the literature in relation to the optimisation of different aspects of the empirical distribution of a parameter ensemble. This includes the constrained Bayes (CB), weighted-rank squared error loss (WRSEL), and triple-goal (GR) ensembles of point estimates. In addition, we also consider the set of maximum likelihood estimates (MLEs) and the ensemble of posterior means –the latter being optimal under the summed squared error loss (SSEL). Firstly, we test the performance of these different sets of point estimates as plug-in estimators for the empirical quantiles and empirical QR under a range of synthetic scenarios encompassing both spatial and non-spatial simulated data sets. Performance evaluation is here conducted using the posterior regret, which corresponds to the difference in posterior losses between the chosen plug-in estimator and the optimal choice under the loss function of interest. The triple-goal plug-in estimator is found to outperform its counterparts and produce close-to-optimal empirical quantiles and empirical QR. A real data set documenting schizophrenia prevalence in an urban area is also used to illustrate the implementation of these methods. Secondly, two threshold classification losses (TCLs) –weighted and unweighted– are formulated. The weighted TCL can be used to emphasise the estimation of false positives over false negatives or the converse. These weighted and unweighted TCLs are optimised by a set of posterior quantiles and a set of posterior medians, respectively. Under an unweighted classification framework, the SSEL point estimates are found to be quasi-optimal for all scenarios studied. In addition, the five candidate plug-in estimators are also evaluated under the rank classification loss (RCL), which has been previously proposed in the literature. The SSEL set of point estimates are again found to constitute quasi-optimal plug-in estimators under this loss function, approximately on a par with the CB and GR sets of point estimates. The threshold and rank classification loss functions are applied to surveillance data reporting methicillin resistant Staphylococcus aureus (MRSA) prevalence in UK hospitals. This application demonstrates that all the studied plug-in classifiers under TCL tend to be more liberal than the optimal estimator. That is, all studied plug-in estimators tended to classify a greater number of hospitals above the risk threshold than the set of posterior medians. In a concluding chapter, we discuss some possible generalisations of the loss functions studied in this thesis, and consider how model specification can be tailored to better serve the inferential goals considered.