  We describe a new paradigm for implementing inference in belief networks,
which relies on compiling a belief network into an arithmetic expression called
a Query DAG (Q-DAG). Each non-leaf node of a Q-DAG represents a numeric
operation, a number, or a symbol for evidence. Each leaf node of a Q-DAG
represents the answer to a network query, that is, the probability of some
event of interest. It appears that Q-DAGs can be generated using any of the
algorithms for exact inference in belief networks --- we show how they can be
generated using clustering and conditioning algorithms. The time and space
complexity of a Q-DAG generation algorithm is no worse than the time complexity
of the inference algorithm on which it is based; that of a Q-DAG on-line
evaluation algorithm is linear in the size of the Q-DAG, and such inference
amounts to a standard evaluation of the arithmetic expression it represents.
The main value of Q-DAGs is in reducing the software and hardware resources
required to utilize belief networks in on-line, real-world applications. The
proposed framework also facilitates the development of on-line inference on
different software and hardware platforms, given the simplicity of the Q-DAG
evaluation algorithm. This paper describes this new paradigm for probabilistic
inference, explaining how it works, its uses, and outlines some of the research
directions that it leads to.
