Bayesian analysis of contingency tables is main-stream statistics. Massam-Letac have given (the only?) careful account of the mathematical details for carrying out conjugate prior analysis for hierarchical models. They managed to pass to the limit of 'zero apriori prior size' and find elegant structure. I will give an account of their work and try to blend it with the tools of partial exchangeability and algebraic statistics. This offers parameter free descriptions of these models and priors in terms of 'symmetries' of the observables and prediction rules. Much remains to be done, but there is some nice preliminary work to report.