Probabilistic Reasoning with Hierarchically Structured Variables

*By Rita Sharma*

Abstract:

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.