Many real life domains contain a mixture of discrete and continuous
variables and can be modeled as *hybrid Bayesian Networks (BNs)*. An
important subclass of hybrid BNs are *conditional linear Gaussian
(CLG)* networks, where the conditional distribution of the continuous
variables given an assignment to the discrete variables is a multivariate
Gaussian. Lauritzen's extension to the clique tree algorithm can be used
for exact inference in CLG networks. However, many domains include discrete
variables that depend on continuous ones, and CLG networks do not allow such
dependencies to be represented. In this paper, we propose the first
"exact" inference algorithm for *augmented CLG* networks --- CLG
networks augmented by allowing discrete children of continuous parents. Our
algorithm is based on Lauritzen's algorithm, and is exact in a similar
sense: it computes the exact distributions over the discrete nodes, and the
exact first and second moments of the continuous ones, *up to inaccuracies
resulting from numerical integration used within the algorithm*.
In the special case of *softmax* CPDs, we show that integration can
often be done efficiently, and that using the first two moments leads to a
particularly accurate approximation. We show empirically that our algorithm
achieves substantially higher accuracy at lower cost than previous
algorithms for this task.