Multi-way Tables with Fixed Marginal Totals"

In this talk I exploit the theory of graphical models to characterize a set of tables W induced by several possibly overlapping marginals. When these marginals are the cliques of a decomposable independence graph, I develop explicit formulas for sharp upper and lower bounds on the cell entries in W. Moreover, I show that simple data swaps or local moves are the only moves required to construct a Markov basis that links all the contingency tables in W. I extend this approach to a divide-and-conquer technique applicable to reducible independence graphs. I introduce the generalized shuttle algorithm that can be used to compute bounds for any configuration of fixed marginals. This algorithm can be modified to exhaustively enumerate all the tables in W and to randomly generate global moves in W as an alternative to Markov bases.

Next I describe approaches to data augmentation in multi-way contingency tables based on local and global moves Metropolis-Hastings simulation as well as on a class of simple and useful prior distributions on the parameters of log-linear models.

I conclude my talk with open issues and future research directions.