In this paper, we establish some theoretical connections between Sum-Product Networks (SPNs) and Bayesian Networks (BNs). We prove that every SPN can be converted into a BN in linear time and space in terms of the network size. The key insight is to use Algebraic Decision Diagrams (ADDs) to compactly represent the local conditional probability distributions at each node in the resulting BN by exploiting context-specific independence (CSI). The generated BN has a simple directed bipartite graphical structure. We show that by applying the Variable Elimination algorithm (VE) to the generated BN with ADD representations, we can recover the original SPN where the SPN can be viewed as a history record or caching of the VE inference process. To help state the proof clearly, we introduce the notion of normal SPN and present a theoretical analysis of the consistency and decomposability properties. We conclude the paper with some discussion of the implications of the proof and establish a connection between the depth of an SPN and a lower bound of the tree-width of its corresponding BN.
@cite_28: We provide a perspective on knowledge compilation which calls for analyzing different compilation approaches according to two key dimensions: the succinctness of the target compilation language, and the class of queries and transformations that the language supports in polytime. We argue that such analysis is necessary for placing new compilation approaches within the context of existing ones. We also go beyond classical, flat target compilation languages based on CNF and DNF, and consider a richer, nested class based on directed acyclic graphs, which we show to include a relatively large number of target compilation languages.
Another representation closely related to SPNs in propositional logic and knowledge representation is the deterministic-Decomposable Negation Normal Form (d-DNNF) @cite_28 . Propositional formulas in d-DNNF are represented by a directed acyclic graph (DAG) structure to enable the re-usability of sub-formulas. The terminal nodes of the DAG are literals and the internal nodes are AND or OR operators. Like SPNs, d-DNNF formulas can be queried to answer satisfiability and model counting problems. We refer interested readers to and for more detailed discussions.