  Log-linear models are a well-established method for describing statistical
dependencies among a set of n random variables. The observed frequencies of the
n-tuples are explained by a joint probability such that its logarithm is a sum
of functions, where each function depends on as few variables as possible. We
obtain for this class a new model selection criterion using nonasymptotic
concepts of statistical learning theory. We calculate the VC dimension for the
class of k-factor log-linear models. In this way we are not only able to select
the model with the appropriate complexity, but obtain also statements on the
reliability of the estimated probability distribution. Furthermore we show that
the selection of the best model among a set of models with the same complexity
can be written as a convex optimization problem.
