A class of smooth models satisfying marginal and context specific conditional independencies

TL;DR

This paper introduces a new class of smooth discrete data models that satisfy marginal and context-specific conditional independencies, addressing non-smooth models by restricting certain independence statements.

Contribution

It proposes a novel marginal log-linear parameterization and a method to restore smoothness through conditional restrictions, supported by a reconstruction algorithm and fixed point theory.

Findings

Non-smooth models are characterized within this class.

Smoothness can be achieved by restricting independence statements.

A general rule for implied conditional independence restrictions is provided.

Abstract

We study a class of conditional independence models for discrete data with the property that one or more log-linear interactions are defined within two different marginal distributions and then constrained to 0; all the conditional independence models which are known to be non smooth belong to this class. We introduce a new marginal log-linear parameterization and show that smoothness may be restored by restricting one or more independence statements to hold conditionally to a restricted subset of the configurations of the conditioning variables. Our results are based on a specific reconstruction algorithm from log-linear parameters to probabilities and fixed point theory. Several examples are examined and a general rule for determining the implied conditional independence restrictions is outlined.