Conditional distributions and log-linear parameters
Robin J. Evans · arXiv (Cornell University) · 2014
Many multivariate statistical models are defined by restrictions on certain marginal or conditional distributions, motivating parameterizations which enable such constraints to be enforced easily. In the discrete case, marginal log-linear parameters are useful for enforcing marginal constraints, and a flexible family of smooth parameterizations using collections of these parameters was introduced by Bergsma and Rudas (2002). However, there are various interesting models defined by combinations of parameters whose smoothness is not yet established. We consider the use of general collections of marginal log-linear parameters to smoothly parameterize joint distributions. Existing results about margins are extended to conditional distributions, and are used to parameterize probability distributions corresponding to systems under causal intervention. We also explore the relationship between log-linear parameters defined within different margins, and use these results to construct iterative methods for recovering joint probability distributions from marginal log-linear pieces. Finally we use Markov chain theory to prove that models defined by certain collections of conditional independences are curved exponential families.