The Bayesian Analysis of Contingency Tables

Summary. This paper describes how data from a multinomial distribution, and in particular data in the form of a contingency table, may be studied by using a prior distribution of the parameters and expressing the results in the form of a posterior distribution, or some aspects thereof, of the parameters. The analysis used must depend on the prior distribution and the form described here only applies to a certain type of prior knowledge but, for reasons given below, it is believed that this type is of frequent occurrence. The binomial situation is first considered and the results obtained there suggest a general result for the multinomial distribution, which is then established. A few remarks on Bayesian analysis in general enable the result to be applied, first to certain multinomial problems and then, with the aid of another general result, to contingency tables. The method used there has close connections with the Analysis of Variance and these connections are examined, particularly with a view to simplifying the analysis of contingency tables involving three or more factors. 1. Binomial distributions. Although it will appear as a special case of results to be established for the general multinomial situation, it is instructive to begin with the binomial distribution which suggested the generalizations. Let N independent trials with constant probability 0 of success result in n successes and (N - n) failures. The likelihood is