Symbolic probabilistic inference with both discrete and continuous variables

The importance of resolving general queries in Bayesian networks using the symbolic probabilistic inference (SPI) algorithm is considered. SPI applies the concept of dependency-directed backward search to probabilistic inference, and is incremental with respect to both queries and observations. Unlike traditional Bayesian network inferencing algorithms, the SPI algorithm is goal directed, performing only those calculations that are required to respond to queries. Research to date on SPI applies to Bayesian networks with only discrete-valued variables or only continuous variables (linear Gaussian) and does not address networks with both discrete and continuous variables. In this paper, we extend the SPI algorithm to handle Bayesian networks made up of both discrete and continuous variables (SPI-DC). The only topological constraint of the networks is that the successors of any continuous variable have to be continuous variables as well. In order to have exact analytical solution, the relationships between the continuous variables are restricted to be “linear Gaussian.” With new representation, SPI-DC modifies the three basic SPI operations: multiplication, summation, and substitution. However, SPI-DC retains the framework of the SPI algorithm, namely building the search tree and recursive query mechanism and therefore retains the goal-directed and incrementality features of SPI