  We provide results demonstrating the smoothness of some marginal log-linear
parameterizations for distributions on multi-way contingency tables. First we
give an analytical relationship between log-linear parameters defined within
different margins, and use this to prove that some parameterizations are
equivalent to ones already known to be smooth. Second we construct an iterative
method for recovering joint probability distributions from marginal log-linear
pieces, and prove its correctness in particular cases. Finally we use Markov
chain theory to prove that certain cyclic conditional parameterizations are
also smooth. These results are applied to show that certain conditional
independence models are curved exponential families.
