Probabilistic Reasoning with Hierarchically Structured Variables

By Rita Sharma


Many practical problems have random variables with a large number of values that can be hierarchically structured into an abstraction tree of classes. This paper considers how to represent and exploit hierarchical structure in probabilistic reasoning. We represent the distribution for such variables by specifying, for each class, the probability distribution over its immediate subclasses. We represent the conditional probability distribution of any variable conditioned on hierarchical variables using inheritance. We present an approach for reasoning in Bayesian networks with hierarchically structured variables that dynamically constructs a flat Bayesian network, given some evidence and a query, by collapsing the hierarchies to include only those values necessary to answer the query. This can be done with a single pass over the network. We can answer the query from the flat Bayesian network using any standard probabilistic inference algorithm such as variable elimination or stochastic simulation. The domain size of the variables in the flat Bayesian network is independent of the size of the hierarchies; it depends on how many of the classes in the hierarchies are directly associated with the evidence and query.
Thus, the representation is applicable even when the hierarchy is conceptually infinite.

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