Inspired by the Dirac notation, a new set of symbols, the Probability Bracket Notation (PBN) is proposed for probability modeling. By applying PBN to discrete and continuous random variables, we show that PBN could play a similar role in probability spaces as the Dirac notation in Hilbert vector spaces. The time evolution of homogeneous Markov chains with discrete-time and continuous-time are discussed in PBN. Our system state p-kets are identified with the probability vectors, while our system state p-bra can be identified with Doi’s state function and Peliti’s standard bra. We also suggest that, by transforming from the Schrodinger picture to the Heisenberg picture, the time-dependence of a system p-ket of a homogeneous MC can be shifted to the observable as a stochastic process. 1.
Inspired by the Dirac notation, a new set of symbols, the Probability Bracket Notation (PBN) is proposed for probability modeling. By applying PBN to discrete and continuous random variables, we show that PBN could play a similar role in probability spaces as the Dirac notation in Hilbert vector spaces. The time evolution of homogeneous Markov chains with discrete-time and continuous-time are discussed in PBN. Our system state p-kets are identified with the probability vectors, while our system state p-bra can be identified with Doi’s state function and Peliti’s standard bra. We also suggest that, by transforming from the Schrodinger picture to the Heisenberg picture, the time-dependence of a system p-ket of a homogeneous MC can be shifted to the observable as a stochastic process. 1.
page1 Abstract Dirac notation has been widely used for vectors in Hilbert spaces of Quantum Theories. In this paper, we propose to use the Probability Bracket Notation (PBN), a new set of symbols defined similarly (but not identically) as in Dirac notation. By applying PBN to fundamental definitions and theorems for discrete and continuous random variables, we show that PBN could play a similar role in probability sample space as Dirac notation in Hilbert vector space. Applying PBN to homogeneous Markov chains (MC) with discrete time, we show that our system state P-kets are identified with the probability vectors in Markov chains (MC). Then we apply PBN to general stochastic processes (SP). The master equation of time-continuous homogeneous MC in the Schrodinger pictures is discussed. Our system state P-bra is identified with Doi's state function and Peliti's standard bra. In the end, we investigated the transition of probability density from the Schrodinger picture to the Heisenberg picture for time-continuous homogeneous MC. We
Dirac notation has been widely used for vectors in Hilbert spaces of Quantum Theories and now also in Information Retrieval. In this paper, we propose to use Probability Bracket Notation (PBN) for probability modeling. The new symbols are defined similarly (but not identically) as in Dirac notation. By using PBN to represent fundamental definitions and theorems for discrete and continuous random variables, we show that PBN could play a similar role in probability sample space as Dirac notation in Hilbert space. We also find a close relation between our system state P-kets and probability vectors in Markov chains. In the end, we apply PBN to some important stochastic processes, present the master equation of time-continuous Markov chains in both Schrodinger and Heisenberg pictures. We identify our system state P-bra with Doi's state function and Peliti's standard bra. We summarize the similarities and differences between PBN and Dirac Notation in the two tables of Appendix A.
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.
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.
After a brief introduction to Probability Bracket Notation (PBN), indicator operator and conditional density operator (CDO), we investigate probability spaces associated with various quantum systems: system with one observable (discrete or continuous), system with two commutative observables (independent or dependent) and a system of indistinguishable non-interacting many-particles. In each case, we derive unified expressions of conditional expectation (CE), conditional probability (CP), and absolute probability (AP): they have the same format for discrete or continuous spectrum; they are defined in both Hilbert space (using Dirac notation) and probability space (using PBN); and they may be useful to deal with CE of non-commutative observables.
In this article, we continue to explore Probability Bracket Notation (PBN), proposed in our previous article. Using both Dirac vector bracket notation (VBN) and PBN, we define induced Hilbert space and induced sample space, and propose that there exists an equivalence relation between a Hilbert space and a sample space constructed from the same base observable(s). Then we investigate Markov transition matrices and their eigenvectors to make diffusion maps with two examples: a simple graph theory example, to serve as a prototype of bidirectional transition operator; a famous text document example in IR literature, to serve as a tutorial of diffusion map in text document space. We show that the sample space of the Markov chain and the Hilbert space spanned by the eigenvectors of the transition matrix are not equivalent. At the end, we apply our PBN and equivalence proposal to Thermophysics by associating sample (phase) space with the Hilbert space of a single particle and the Fock space of many-particle systems.