A first approach to closeness distributions

We start by introducing a simple example to illustrate the kind of problems we are interested in solving. Consider the problem of estimating a parameter θ using data from a small experiment and a prior distribution constructed from similar previous experiments. The specific problem description is borrowed from [3]: Estimating the risk of tumor in a group of rats. In the evaluation of drugs for possible clinical application, studies are routinely performed on rodents. For a particular study drawn from the statistical literature, suppose the immediate aim is to estimate θ, the probability of tumor in a population of female laboratory rats of type'F344' that receive a zero dose of the drug (a control group). The data show that 4 out of 14 rats developed endometrial stromal polyps (a kind of tumor). Typically, the mean and standard deviation of underlying tumor risks are not available. Rather, historical data are available on previous experiments on similar groups of rats. In the rat tumor example, the historical data were in fact a set of observations of tumor incidence in 70 groups of rats (table 1).