Bayes factors for independence in contingency tables

SUMMARY The null hypothesis of row-column independence in a two-way contingency table can be expressed as a constraint on the parameters in various standard statistical sampling models. Four sampling models are considered, which are related by nested conditioning. By having the prior distribution in any one model induce the prior distribution in each further conditioned model, it is shown that the Bayes factors for independence will factorize, and thereby expose the evidence residing in the marginal row and column of the table. Bounds on the marginal Bayes factors justify, in a weak sense, Fisher's practice of conditioning. A general theorem is given for factorized Bayes factors from a factorized likelihood function.