Graphical Markov Models: Overview

We describe how graphical Markov models emerged in the last 40 years, based on three essential concepts that had been developed independently more than a century ago. Sequences of joint or single regressions and their regression graphs are singled out as being the subclass that is best suited for analyzing longitudinal data and for tracing developmental pathways, both in observational and in intervention studies. Interpretations are illustrated using two sets of data. Furthermore, some of the more recent, important results for sequences of regressions are summarized. 1 Some general and historical remarks on the types of model Graphical models aim to describe in concise form the possibly complex interrelations between a set of variables so that key properties can be read directly o↵ a graph. The central idea is that each variable is represented by a node in a graph. Any pair of nodes may become coupled, that is joined by an edge. Coupled nodes are also said to be adjacent. For many types of graph, a missing edge represents some form of conditional independence between the pair of variables and an edge present can be interpreted as a corresponding conditional dependence. Because the conditioning set may be empty, or may contain some or all of the other variables, a variety of types of graph have been developed and are used to represent di↵erent types of structure. A particularly important distinction is between directed and undirected edges. In the former an arrow indicates the direction of dependence of a response on an explanatory variable, the latter is also called a regressor. If, on the other hand, two variables are to be interpreted on an equal standing then the edge between them is typically undirected.