arXiv:2402.15075v1 Announce Type: new

Abstract: Hybrid Bayesian networks (HBN) contain complex conditional probabilistic distributions (CPD) specified as partitioned expressions over discrete and continuous variables. The size of these CPDs grows exponentially with the number of parent nodes when using discrete inference, resulting in significant inefficiency. Normally, an effective way to reduce the CPD size is to use a binary factorization (BF) algorithm to decompose the statistical or arithmetic functions in the CPD by factorizing the number of connected parent nodes to sets of size two. However, the BF algorithm was not designed to handle partitioned expressions. Hence, we propose a new algorithm called stacking factorization (SF) to decompose the partitioned expressions. The SF algorithm creates intermediate nodes to incrementally reconstruct the densities in the original partitioned expression, allowing no more than two continuous parent nodes to be connected to each child node in the resulting HBN. SF can be either used independently or combined with the BF algorithm. We show that the SF+BF algorithm significantly reduces the CPD size and contributes to lowering the tree-width of a model, thus improving efficiency.

Hybrid Bayesian networks (HBN) are a powerful tool for modeling and reasoning under uncertainty in situations where the variables of interest involve both discrete and continuous domains. However, the complexity of the conditional probabilistic distributions (CPDs) in HBNs can grow exponentially with the number of parent nodes when using discrete inference, resulting in inefficiency.

To address this issue, researchers have developed various techniques to reduce the size of CPDs. One common approach is binary factorization (BF), where the statistical or arithmetic functions in the CPD are decomposed into sets of size two. This can effectively reduce the size of CPDs, but it cannot handle partitioned expressions, limiting its applicability in HBNs.

In this article, the authors propose a new algorithm called stacking factorization (SF) to address the limitations of BF in handling partitioned expressions. The SF algorithm introduces intermediate nodes to incrementally reconstruct the densities in the original partitioned expression. By allowing no more than two continuous parent nodes to be connected to each child node in the resulting HBN, SF overcomes the limitations of BF.

The SF algorithm can be used independently or combined with the BF algorithm. The authors demonstrate that their proposed SF+BF algorithm significantly reduces the size of CPDs and also contributes to lowering the tree-width of a model. This improvement in efficiency can have practical implications, as it allows for more efficient inference and reasoning in HBNs.

The concepts discussed in this article highlight the multi-disciplinary nature of HBNs. On one hand, they involve concepts from probability theory and statistics, such as conditional probabilities and factorization algorithms. On the other hand, they also require an understanding of computational techniques and algorithms, as efficiency and scalability are important considerations when dealing with large-scale HBN models.

Going forward, it would be interesting to see how the SF+BF algorithm performs on real-world applications. Additionally, further research could explore the combination of SF with other techniques for CPD factorization, as well as its potential applications in domains beyond HBNs.