Probability Bracket Notation, Probability Vectors, Markov Chains and Stochastic Processes

Abstract: 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.