The standard statistical analysis of data classified in two ways (say into rows and columns) is through an analysis of variance that splits the total variation of the data into the main effect of rows, the main effect of columns, and the interaction between rows and columns. This paper presents an alternative Bayesian analysis of the same situation that is appropriate for certain types of prior knowledge. It leads to a rather different treatment of the three factors just mentioned. This analysis proposes a four stage model. The first stage describes the dependence of the x's on the thetas; the second, that of the thetas on the alphas and betas; the third describes the structure of the alphas and betas; and a fourth stage is necessary to describe the prior distributions of mus. Appendices provide derivations and a numerical example. (Author/CTM)