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Probability Bracket Notation: Multivariable Systems and Static Bayesian Networks

The Probability Bracket Notation (PBN) is used to analyze multiple discrete random variables in static Bayesian Networks (BN) through probabilistic graphical models. We briefly introduce the definitions of probability distributions in multivariable systems and their presentations using PBN, then explore the well-known student BN. Our analysis includes calculating various joint, marginal, intermediate, and conditional probability distributions, completing homework assignments, examining relationships between variables (dependence, independence, and conditional independence), and disclosing the power of and restrictions on inserting P-identity operators. We also show the reasoning capabilities of the Student BN using bottom-up and top-down approaches, validated by Elvira software. In the last section, we discuss BNs with continuous variables. After reviewing linear Gaussian networks, we introduce a customized Healthcare BN that includes continuous and discrete random variables, incorporates user-specific data, and offers tailored predictions through discrete-display (DD) nodes, serving as proxies for their continuous variable parents. Our investigation demonstrates that the PBN delivers a reliable and efficient approach for managing multiple variables in static Bayesian networks, a crucial aspect of Machine Learning (ML) and Artificial Intelligence (AI).

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