Probability Bracket Notation for Probability Modeling

TL;DR

This paper introduces a probability bracket notation (PBN) inspired by quantum mechanics' Dirac notation, enabling abstract, basis-independent probability expressions and simplifying the formulation of stochastic processes and Markov dynamics.

Contribution

The paper presents a novel probability bracket notation (PBN) that generalizes quantum Dirac notation for probability modeling, facilitating easier formulation and expansion of probabilistic systems.

Findings

PBN allows basis-independent probability expressions.

Time evolution of Markov processes is elegantly represented.

Quantum expressions are transformed into probability expressions via Wick rotation.

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.