Bayesian Networks

Bayesian networks -- also known as "belief networks" or "causal networks" -- are graphical models for representing multivariate probability distributions. Each variable tex2html_wrap_inline72 is represented as a vertex in an directed acyclic graph ("dag"); the probability distribution tex2html_wrap_inline74 is represented in factorized form as follows:


where tex2html_wrap_inline76 is the set of vertices that are tex2html_wrap_inline78 's parents in the graph. A Bayesian network is fully specified by the combination of:

A small example Bayesian network structure for a (somewhat facetious/futuristic) medical diagnostic domain is shown below. This network might be used to diagnose whether a patient is suffering from a mere common cold (C) and/or the more dangerous Martian Death Flu (M), based on the patients' symptoms -- whether or not the patient has a runny nose (R), whether or not the patient has a headache (H), and whether or not the patient occasionally spontaneously bursts into flames (S) -- as well as relevant background information, namely whether or not he or she has previously visited Mars (V).


Assuming all six variables are binary, with 1 representing ``true'' and 0 ``false'', the probability tables for the network might be defined as follows:

tabular17 tabular24 tabular31 tabular38 tabular45 tabular52

Once a Bayesian network has been specified, it may be used to compute any conditional probability one wishes to compute. For example, given that a person has recently visited Mars and has a runny nose, the network above could be used to compute the probability tex2html_wrap_inline96 that the person has the common cold but not the Martian Death Flu.

Bayesian networks are very convenient for representing systems of probabilistic causal relationships. The fact ``X often causes Y'' may easily be modeled in the network by adding a directed arc from X to Y and setting the probabilities appropriately. On the other hand, if A has no causal influence on B, we may simply leave out an arc from A to B. (For example, there is no arc from C to S in the network above, since the common cold presumably neither causes nor prevents spontaneous combustion.)

Some important Bayesian network caveats / research areas :

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