Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Probability Bracket Notation, Probability Vectors, Markov Chains and
  Stochastic Processes   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.

32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin