Research on Symbolic Probabilistic Inference (SPI) [2, 3] has provided an algorithm for resolving general queries in Bayesian networks. 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, 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 discrete-valued variables and does not address variables with continuous values. In this papers, we extend the SPI algorithm to handle Bayesian networks made up of continuous variables where the relationships between the variables are restricted to be ?linear gaussian?. We call this variation of the SPI algorithm, SPI Continuous (SPIC). SPIC modifies the three basic SPI operations: multiplication, summation, and substitution. However, SPIC 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.

Recent research on the Symbolic Probabilistic Inference (SPI) algorithm[2] has focused attention on the importance of resolving general queries in Bayesian networks. SPI applies the concept of dependency-directed backward search to probabilistic inference, and is incremental with respect to both queries and observations. In response to this research we have extended the evidence potential algorithm [3] with the same features. We call the extension symbolic evidence potential inference (SEPI). SEPI like SPI can handle generic queries and is incremental with respect to queries and observations. While in SPI, operations are done on a search tree constructed from the nodes of the original network, in SEPI, a clique-tree structure obtained from the evidence potential algorithm [3] is the basic framework for recursive query processing. In this paper, we describe the systematic query and caching procedure of SEPI. SEPI begins with finding a clique tree from a Bayesian network-the standard procedure of the evidence potential algorithm. With the clique tree, various probability distributions are computed and stored in each clique. This is the ?pre-processing? step of SEPI. Once this step is done, the query can then be computed. To process a query, a recursive process similar to the SPI algorithm is used. The queries are directed to the root clique and decomposed into queries for the clique's subtrees until a particular query can be answered at the clique at which it is directed. The algorithm and the computation are simple. The SEPI algorithm will be presented in this paper along with several examples.

Díez's algorithm for the noisy MAX is very efficient for polytrees, but when the network has loops, it has to be combined with local conditioning, a suboptimal propagation algorithm. Other algorithms, based on several factorizations of the conditional probability of the noisy MAX, are not as efficient for polytrees but can be combined with general propagation algorithms such as clustering or variable elimination, which are more efficient for networks with loops. In this article we propose a new factorization of the noisy MAX that amounts to Díez's algorithm in the case of polytrees and at the same time is more efficient than previous factorizations when combined with either variable elimination or clustering.