A general approach is proposed for modeling the structure of a two-way contingency table, and for drawing inferences about the marginal and interaction effects, cell parameters, and conditional probabilities. The prior distribution expresses uncertainty in a simple reduced model, in particular the independence model. The posterior estimates of the cell parameters then provide compromises between the cell frequencies and fitted values obtained under the reduced model. In a mental test context, the reduced independence model is identical to Rasch's multiplicative Poisson model, and we therefore incorporate a procedure for checking the adequacy of this model. The use of some general ideas on marginalization makes it possible to computer reasonable approximations to the full posterior densities of many parameters, thus permitting thorough parametric inference and statistical modeling. All parameters are estimated via a hierarchical Bayes procedure. A cross classification of 5648 Marine Corps clerical students, by school and test grade, is analyzed in detail. The posterior densities of the 96 possible interactions suggest a simplified structure partitioning and collapsing of the table into a meaningful 3 x2 table. (Author/GDC)