ABSTRACT
A simple two-component mixture density f takes the form f( y I 8) 7Tft(Y I OJ) + (l - 1r)f2(Y I 82) for densities ft, f2 and 0 s 7T :s 1. Given a sample of size n from such a density, the likelihood is effectively the sum of 2n component terms, and the implementation of Bayesian methods runs up against considerable computational difficulties. Recently, however, it has been shown that the Gibbs sampler can be used to remove this problem via the introduction of latent variables, thus allowing fully Bayesian inference to be performed where previously this was not feasible. In this chapter we study a version of the two-component mixture model with particular reference to the following problem in medical statistics. Low birth weight is widely considered to be an important factor in perinatal mortality, but it is also recognized that it is not sufficient to take action on the basis of the birth weight figure alone. It is also necessary
to distinguish between values in the lower tail of the predominant (lowrisk) population and values derived from a separate, residual (high-risk) group. Thus, the probability distribution of birth weights is best modeled as a two-component mixture. For our purposes in this chapter, the densities can be adequately modeled as normal, with the residual density somewhat more heavy-tailed than the predominant density. The clinical objective is to identify a thresholding criterion to aid classification of individuals into the low/high-risk groups. Novel features include the treatment of rounding and the implicitly "rounded integer" form of measurements.
