The clique tree algorithm is the standard method for doing inference in
Bayesian networks.  It works by manipulating <I>clique potentials</I> ---
distributions over the variables in a clique.  While this approach works
well for many networks, it is limited by the need to maintain an exact
representation of the clique potentials.  This makes the clique tree
algorithm impractical for certain large discrete networks, and inapplicable
for most continuous and hybrid networks.  This paper presents a new unified
approach that combines approximate inference and the clique tree algorithm,
thereby circumventing this limitation.  Many known approximate inference
algorithms can be viewed as instances of this approach: likelihood
weighting, most approximate inference algorithms for dynamic Bayesian
networks, inference for Conditional Gaussian networks, and more.  The
algorithm essentially does clique tree propagation, using approximate
inference to estimate the densities in each clique.  In many settings, the
computation of the approximate clique potential can be done easily using
statistical importance sampling.  Unlike exact clique tree propagation, the
results of a single pass may not be accurate enough.  The algorithm deals
with this problem using an iterative scheme: it estimates the error at each
clique, and re-evaluates the potential of those where the error is large.
This algorithm combines the best features of exact and approximate
inference: Like approximate inference algorithms, it can deal with complex
domains, including hybrid networks.  Like the clique tree algorithm, it
exploits the locality structure of the Bayes net to reduce the
dimensionality of the probability densities involved in the computation.
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