Probability Bracket Notation for Probability Modeling

Axioms, vol. 13, pp. 564

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 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 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.

Authors 2

1. Xing Min Wang
   Affiliation as printed

   Sherman Visual Lab, Sunnyvale, CA 94085, USA

2. Tony C. Scott corresponding Aachen Institut für Physikalische Chemie

   RWTH Aachen University

   Affiliation as printed

   Institut für Physikalische Chemie, RWTH Aachen University, 52056 Aachen, Germany