We provide results demonstrating the smoothness of some marginal log-linearparameterizations for distributions on multi-way contingency tables. First wegive an analytical relationship between log-linear parameters defined withindifferent margins, and use this to prove that some parameterizations areequivalent to ones already known to be smooth. Second we construct an iterativemethod for recovering joint probability distributions from marginal log-linearpieces, and prove its correctness in particular cases. Finally we use Markovchain theory to prove that certain cyclic conditional parameterizations arealso smooth. These results are applied to show that certain conditionalindependence models are curved exponential families.
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