Finite mixture models
Mixture distributions arise in practical problems when the measurements of a random variable are taken under two different conditions. For example, the distribution of heights in a population of adults reflects the mixture of males and females in the population, and the reaction times of schizophrenics on an attentional task might be a mixture of trials in which they are or are not affected by an attentional delay (an example discussed later in this chapter). For the greatest flexibility, and consistent with our general hierarchical modeling strategy, we construct such distributions as mixtures of simpler forms. For example, it is best to model male and female heights as separate univariate, perhaps normal, distributions, rather than a single bimodal distribution. This follows our general principle of using conditioning to construct realistic probability models. The schizophrenic reaction times cannot be handled in the same way because it is not possible to identify which trials are affected by the attentional delay. Mixture models can be used in problems of this type, where the population of sampling units consists of a number of subpopulations within each of which a relatively simple model applies. In this chapter we discuss methods for analyzing data using mixture models.