ABSTRACT

Many statistical applications involve multiple parameters that can be regarded as related or connected in some way by the structure of the problem, implying that a joint probability model for these parameters should reflect the dependence among them. For example, in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability θj, it might be reasonable to expect that estimates of the θj’s, which represent a sample of hospitals, should be related to each other. We shall see that this is achieved in a natural way if we use a prior distribution in which the θj’s are viewed as a sample from a common population distribution. A key feature of such applications is that the observed data, yij, with units indexed by i within groups indexed by j, can be used to estimate aspects of the population distribution of the θj’s even though the values of θj are not themselves observed. It is natural to model such a problem hierarchically, with observable outcomes modeled conditionally on certain parameters, which themselves are given a probabilistic specification in terms of further parameters, known as hyperparameters. Such hierarchical thinking helps in understanding multiparameter problems and also plays an important role in developing computational strategies.