Xing M. Wang
From Dirac Notation to Probability Bracket Notation: Term Vector Space, Concept Fock Space and Probabilistic IR Models.
Tue, 03/22/2011 - 16:01 — SomNambulAuthors: Xing M. Wang
Subjects: Information Retrieval
AbstractAfter a brief introduction to Probability Bracket Notation (PBN) for discrete
random variables in time-independent probability spaces, we apply both PBN and
Dirac notation to investigate probabilistic modeling for information retrieval
(IR). We derive the ranking formulas for various probabilistic models, induced
by Term Vector Space (TVS) and by Concept Fock Space (CFS). The ranking
formulas are naturally expressed in term frequencies; and, because our formulas
for inference network models (INM) are symmetric, they can also be used to rank
closeness of documents.Probability Bracket Notation: the Unified Expressions of Conditional Expectation and Conditional Probability in Quantum Modeling.
Tue, 11/10/2009 - 16:01 — SomNambulAuthors: Xing M. Wang
Subjects: Probability
AbstractAfter 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 dProbability Bracket Notation: Probability Space, Conditional Expectation and Introductory Martingales.
Fri, 10/16/2009 - 17:25 — SomNambulAuthors: Xing M. Wang
Subjects: Probability
AbstractIn this paper, we continue to explore the consistence and usability of
Probability Bracket Notation (PBN) proposed in our previous articles. After a
brief review of PBN with dimensional analysis, we investigate probability
spaces in terms of PBN by introducing probability spaces associated with random
variables (R.V) or associated with stochastic processes (S.P). Next, we express
several important properties of conditional expectation (CE) and some their
proofs in PBN. Then, we introduce martingales based on sequence of R.V or based
on filtration in PBN.Probability Bracket Notation and Probability Modeling.
Tue, 09/08/2009 - 12:16 — SomNambulAuthors: Xing M. Wang
Subjects: Probability
AbstractInspired 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 the Doi state function or the Peliti standard bra.Probability Bracket Notation and Probability Modeling.
Tue, 09/08/2009 - 12:16 — SomNambulAuthors: Xing M. Wang
Subjects: Probability
AbstractInspired 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 the Doi state function or the Peliti standard bra.
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