📝 SummarXiv

Abstract

We expand the Probability Bracket Notation (PBN), a symbolic framework inspired by the Dirac notation in quantum mechanics, to multivariable probability systems and static Bayesian networks (BNs). By introducing PBN for joint, marginal, and conditional probability distributions (PDs), as well as marginal and conditional expectations, we demonstrate how to express dependencies among multiple random variables concisely and manipulate them algebraically. Using the well-known Student BN as an example of probabilistic graphical models (PGMs), we show how to apply PBN to analyze predictions, inferences (using both bottom-up and top-down approaches), and expectations. We also extend PBN to BNs with continuous variables. After reviewing linear Gaussian networks, we introduce a customized Healthcare BN that includes both continuous and discrete random variables, utilizes user-specific data, and provides tailored predictions through discrete-display (DD) nodes as proxies for their continuous variable parents. Compared to traditional probability notation, PBN offers a unifying operator-like framework that simplifies the analysis of probabilistic models. This work highlights the potential of PBN as both an educational tool and a practical framework for probabilistic modeling, paving the way for applications in causal reasoning, inferences, expectations, data analytics, machine learning, and artificial intelligence.