A Brief Introduction to Graphical Models and How to Learn Them from Data

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Nov 7, 2013 (3 years and 11 months ago)

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A Brief Introduction to Graphical Models


and How to Learn Them from Data



Christian Borgelt


Dept. of Knowledge Processing and Language Engineering

School of Computer Science

Otto
-
von
-
Guericke
-
University of Magdeburg

Universitätsplatz 2, 39106 Magdeburg,

Germany

borgelt@iws.cs.uni
-
magdeburg.de

http://fuzzy.cs.uni
-
magdeburg.de/~borgelt/



Abstract


In the last decade probabilistic graphical models
-

in particular

Bayes networks and Markov networks
-

bec
ame very popular as tools

for structuring uncertain knowledge about a domain of interest and

for building knowledge
-
based systems that allow sound and efficient

inferences about this domain. The core idea of graphical models is

that usually certain indepen
dence relations hold between the attributes

that are used to describe a domain of interest. In most uncertainty

calculi
--

and in particular in probability theory
--

the structure of

these independence relations is very similar to properties concerning

the

connectivity of nodes in a graph. As a consequence, it is tried

to capture the independence relations by a graph, in which each node

represents an attribute and each edge a direct dependence between

attributes. In addition, provided that the graph capture
s only valid

independences, it prescribes how a probability distribution on the

(usually high
-
dimensional) space that is spanned by the attributes

can be decomposed into a set of smaller (marginal or conditional)

distributions. This decomposition can be ex
ploited to derive evidence

propagation methods and thus enables sound and efficient reasoning

under uncertainty. The lecture gives a brief introduction into the

core ideas underlying graphical models, starting from their relational

counterparts and highlig
hting the relation between independence and

decomposition. Furthermore, the basics of model construction and

evidence propagation are discussed, with an emphasis on join/junction

tree propagation. A substantial part of the lecture is then devoted

to learni
ng graphical models from data, in which quantitative learning

(parameter estimation) as well as the more complex qualitative or

structural learning (model selection) are studied. The lecture closes

with a brief discussion of example applications.



Referen
ces


C. Borgelt and R. Kruse. Graphical Models
-

Methods for Data Analysis and
Mining. J. Wiley & Sons, Chichester, United Kingdom 2002


E. Castillo, J.M. Gutierrez, and A.S. Hadi. Expert Systems and Probabilistic
Network Models. Springer
-
Verlag, New York
, NY, USA 1997


F.V. Jensen. Bayesian Networks and Decision Graphs. Springer
-
Verlag,
New York, NY, USA 2001


S.L. Lauritzen. Graphical Models. Oxford University Press, Oxford, United
Kingdom 1996


R. Neapolitan.
Learning Bayesian Networks.

Prentice Hall, U
pper Saddle
River, NJ, USA 2003


J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of
Plausible Inference (2nd edition). Morgan Kaufman, San Mateo, CA, USA
1992


J. Whittaker. Graphical Models in Applied Multivariate Statistics. J. Wiley
& Sons, Chichester, United Kingdom 1990


Keywords


Graphical model, Bayes network, Markov network, conditional
independence, learning from data