Probabilistic graphical models based on imprecise probability
Credal networks are probabilistic graphical models based on imprecise probability. Credal networks can be regarded as an extension of Bayesian networks, where credal sets replace probability mass functions in the specification of the local models for the network variables given their parents. As a Bayesian network defines a joint probability mass function over its variables, a credal network defines a joint credal set. The way this credal set is defined depends on the particular notion of independence for imprecise probability adopted. Most of the research on credal networks focused on the case of strong independence. Given strong independence the joint credal set associated to a credal network is called its strong extension. Let formula_1 denote a collection of categorical variables and formula_2. If formula_3 is, for each formula_4, a conditional credal set over formula_5, then the strong extension of a credal network is defined as follows:
 formula_6
where formula_7 denote the convex hull.
Inference.
Inference on a credal network is intended as the computation of the bounds of an expectation with respect to its strong extensions. When computing the bounds of a conditional event, inference is called updating. Say that the queried variable formula_8 and the observed variables are formula_9, the lower bound to be evaluated is:
 formula_10
Being a generalization of the same problem for Bayesian networks, updating with credal networks is a NP-hard task. Yet a number of algorithm have been specified.
References.
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