Probability Bracket Notation for Probability Modeling

• URL: http://arxiv.org/abs/2502.04257v1
• Date: Thu, 06 Feb 2025 17:42:53 GMT
• Title: Probability Bracket Notation for Probability Modeling
• Authors: Xing M. Wang, Tony C. Scott,
• Abstract summary: We propose the Bracket Notation (PBN) by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc.<n>Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Following the Dirac Notation in Quantum Mechanics (QM), we propose the Bracket Notation (PBN) by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are 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 system P-bra is comparable to the Doi state function or the Peliti 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 the Dirac notation are naturally transformed into 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.

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