The Bayes factor is a popular method of model selection that compares

the posterior probabilities of two competing models. Consider data given in

the form of a contingency table where N objects are classified according to q

random variables and the conditional independence structure of these random

variables are represented by a discrete graphical model. We assume the cell

counts follow a multinomial distribution with a hyper Dirichlet prior distribution

imposed on the cell probability parameters.We examine the behaviour of

the Bayes factor when the dimension of the model is fixed and when the dimension

increases to infinity with the sample size. Our main result is proving

strong model selection consistency for increasing dimension both when the

true graph is decomposable and when the true graph is non-decomposable.

When the true graph is non-decomposable, we prove that the Bayes factor

selects a minimal triangulation of the true graph.