Applied Bayesian Modelling, 2nd Edition

Chapter 2Hierarchical models for related units

2.1 Introduction: Smoothing to the hyper population

A relatively simple Bayesian problem but one which has motivated much research is that of ensemble estimation, namely estimating the parameters of a common distribution thought to underlie a collection of indicators for similar types of units, sometimes called a hyper-population. Among possible examples are medical, sports, or educational: exam success rates in different schools, results from a set of randomised controlled trials, surgical mortality rates in a set of hospitals (Austin et al., 2001; Austin, 2002), or a run of batting averages for baseball players. A histogram plot of such indicators will typically be ragged, but often appear to suggest an underlying smoother density. While point estimates (e.g. based on fixed effects maximum likelihood) are generally considered unbiased, they are subject to variance instability and may provide unreliable inferences from comparisons, and other properties (e.g. improved stability and precision) may be relevant (Greenland, 2000). While technically unbiased, fixed effects estimators are also subject to measurement or design errors.

So underlying the unit-specific estimates one may posit an underlying set of latent effects characterized by a common density. The hyperparameters of this density are themselves unknowns, and generally include an average effect size and variance, the latter representing heterogeneity in latent outcomes between ...