Probability Bracket Notation: Multivariable Systems and Static Bayesian Networks
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arXiv
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| Format: | Preprint |
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2012
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| _version_ | 1866910203585232896 |
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| author | Wang, Xing M. |
| author_facet | Wang, Xing M. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1207_5293 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Probability Bracket Notation: Multivariable Systems and Static Bayesian Networks Wang, Xing M. Artificial Intelligence Probability 62F15 G.3; I.2.3 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. |
| title | Probability Bracket Notation: Multivariable Systems and Static Bayesian Networks |
| topic | Artificial Intelligence Probability 62F15 G.3; I.2.3 |
| url | https://arxiv.org/abs/1207.5293 |