A central aspect of statistical science is the assessment of
dependences among a set of stochastic variables. The familiar concepts
of correlation, regression, and prediction are manifestations of this
idea, and many aspects of causal relationships ultimately rest on
representations of multivariate dependence.
Graphical Markov models (GMM) use graphs, either undirected,
directed, or mixed, to represent multivariate dependences in a visual
and computationally efficient manner. By representing each variable as
a node in a graph a GMM is usually constructed by specifying local
dependences for each node of the graph in terms of its immediate
neighbors, parents, or both. A GMM can thus represent a highly varied
and complex system of multivariate dependences by means of the global
structure of the graph. The local specification permits efficiencies
in modeling, inference, and probabilistic calculations.
For a fixed graph  model, the classical methods of statistical
inference may be utilized. In many applied domains, however, such as
expert systems for medical diagnosis, weather forecasting, or the
analysis of gene-expression data, the graph is unknown and is itself
the first goal of the analysis. This poses numerous challenges,
including the following:
• The numbers of possible graphs and models grow
superexponentially in the number of variables.
• Distinct graphs G may be Markov equivalent  statistically
indistinguishable.
• Conversely, the same graph may possess different Markov
interpretations.
Furthermore, in applications, GMMs represent one of the most
interdisciplinary topics of contemporary statistical science.
Applications arise in a host of areas, e.g., computer science (expert
systems, robotics, data-mining, machine learning), electrical
engineering (automatic speech recognition systems, error-correcting
codes), genetics (modelling gene-expression data), epidemiology
(causal models), econometrics (structural equations), and behavioral
science (modelling social networks).

References:
Cox,D.R. and Wermuth, N. (1996) Multivariate Dependencies: Models,
Analysis, and Interpretation. Chapman and Hall, London.

Edwards, D. (2000). Introduction to Graphical Modeling, 2nd ed.
Springer, New York.

Lauritzen, S.L. (1996) Graphical models. Oxford University Press, Oxford.

Whittaker, J.L. (1990) Graphical models in Applied Multivariate
Statistics. Wiley, New York.