Probability Bracket Notation: Multivariable Systems and Static Bayesian Networks

TL;DR

This paper introduces Probability Bracket Notation (PBN), inspired by quantum mechanics, to unify and simplify probabilistic reasoning in multivariable systems and Bayesian networks, improving computational efficiency.

Contribution

The paper extends PBN to multivariable systems and Bayesian networks, demonstrating a unified algebraic framework and efficient inference methods, including for hybrid discrete-continuous models.

Findings

Inference along a d-separable chain requires O(k2^k) operations after preprocessing.

PBN can handle continuous variables, including linear Gaussian models.

A hybrid Healthcare BN with discrete display nodes enables user-specific predictions.

Abstract

We extend Probability Bracket Notation (PBN), inspired by the Dirac notation in quantum mechanics, to multivariable probability systems and static Bayesian networks (BNs). By defining probability distributions and conditional expectations in a unified, basis-independent algebraic form, PBN provides a systematic way to represent and manipulate dependencies among random variables. Using the well-known Student BN as an illustrative probabilistic graphical model, we demonstrate prediction, bottom-up and top-down inference, and expectation calculations within the PBN framework. We show that, for a large N-node binary BN, after a one-time preprocessing, inference along a d-separable chain with k intermediate nodes requires O(k2^k) operations, compared to O(N2^N) for direct computation from the full joint distribution. We further extend PBN to networks with continuous variables, including linear Gaussian models, and introduce a hybrid Healthcare BN that combines discrete and continuous variables. In this model, discrete-display nodes serve as proxies for continuous parents, enabling user-specific predictions. Overall, PBN provides an operator-based framework that unifies representation and computation, with potential applications in education, data analytics, and machine learning.