Smoothness of marginal log-linear parameterizations

TL;DR

This paper investigates the smoothness of marginal log-linear parameterizations in multi-way contingency tables, providing analytical relationships, iterative recovery methods, and Markov chain-based proofs to establish their properties.

Contribution

It introduces new analytical and iterative methods to demonstrate the smoothness of various marginal log-linear parameterizations and their applications to conditional independence models.

Findings

Certain marginal log-linear parameterizations are proven to be smooth.

An iterative method for recovering joint distributions is validated.

Conditional independence models are shown to be curved exponential families.

Abstract

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.