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Probability Bracket Notation for Probability Modeling
Version 1
: Received: 22 April 2024 / Approved: 22 April 2024 / Online: 22 April 2024 (11:16:21 CEST)
How to cite: Wang, X. M.; Scott, T. C. Probability Bracket Notation for Probability Modeling. Preprints 2024, 2024041399. https://doi.org/10.20944/preprints202404.1399.v1 Wang, X. M.; Scott, T. C. Probability Bracket Notation for Probability Modeling. Preprints 2024, 2024041399. https://doi.org/10.20944/preprints202404.1399.v1
Abstract
Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc…Using the PBN, many formulas, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman-Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential Applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.
Keywords
Bra-ket notation; Chapman-Kolmogorov Theorem; Information Technology; Markov processes; Quantum Mechanics; Wick rotation
Subject
Physical Sciences, Quantum Science and Technology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Herein, we propose a Probability Bracket Notation (PBN) to play as a similar
tool for the probability modeling. It is found useful to be useful
to build a very useful bridge between classical and quantum worlds
(through special Wick rotation) and provide the possibility to treat
non-Hermitian Hamilton operators.