Discrete chain graph models

TL;DR

This paper examines different Markov properties for discrete chain graph models, showing that only the models analogous to multivariate regressions are smooth, with implications for their statistical properties.

Contribution

It analyzes the structural properties of discrete chain graph models under various Markov properties, identifying which classes are smooth or non-smooth.

Findings

Two Markov property classes can produce non-smooth models.

The class analogous to multivariate regressions is always smooth.

Likelihood functions are unimodal for complete chain components.

Abstract

The statistical literature discusses different types of Markov properties for chain graphs that lead to four possible classes of chain graph Markov models. The different models are rather well understood when the observations are continuous and multivariate normal, and it is also known that one model class, referred to as models of LWF (Lauritzen--Wermuth--Frydenberg) or block concentration type, yields discrete models for categorical data that are smooth. This paper considers the structural properties of the discrete models based on the three alternative Markov properties. It is shown by example that two of the alternative Markov properties can lead to non-smooth models. The remaining model class, which can be viewed as a discrete version of multivariate regressions, is proven to comprise only smooth models. The proof employs a simple change of coordinates that also reveals that the model's likelihood function is unimodal if the chain components of the graph are complete sets.