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Erschienen in:
2013 | OriginalPaper | Buchkapitel
3. Probabilities
verfasst von : Uffe B. Kjærulff, Anders L. Madsen
Erschienen in: Bayesian Networks and Influence Diagrams: A Guide to Construction and Analysis
Verlag: Springer New York
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Abstract
The fact that the structure of a probabilistic network can be characterized as a DAG derives from basic axioms of probability calculus leading to recursive factorization of a joint probability distribution into a product of lower-dimensional conditional probability distributions. First, any joint probability distribution can be decomposed (or factorized) into a product of conditional distributions of different dimensionality, where the dimensionality of the largest distribution is identical to the dimensionality of the joint distribution. This gives rise to a densely connected DAG. Second, statements of local conditional independences manifest themselves as reductions of dimensionalities of some of the conditional probability distributions. Most often, these independence statements give rise to dramatic reductions of complexity of the DAG such that the resulting DAG appears to be quite sparse.