Induced Hilbert Space, Markov Chain, Diffusion Map and Fock Space in Thermophysics

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.Comment: 25 page

Probability Bracket Notation, Term Vector Space, Concept Fock Space and Induced Probabilistic IR Models

After 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 expressions of relevance of document to query (RDQ) for various probabilistic models, induced by Term Vector Space (TVS) and by Concept Fock Space (CFS). The inference network model (INM) formula is symmetric and can be used to evaluate relevance of document to document (RDD); the CFS-induced models contain ingredients of all three classical IR models. The relevance formulas are tested and compared on different scenarios against a famous textbook example.Comment: 23 pages; added a simple example of Bayesian inference; added more test scenarios (e.g., weight formulas); added more reference

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.Comment: 36 pages; two table

Probability Bracket Notation: Probability Space, Conditional Expectation and Introductory Martingales

In 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. In the process, we see PBN can be used to investigate some probability problems, which otherwise might need explicit usage of Measure theory. Whenever applicable, we use dimensional analysis to validate our formulas and use graphs for visualization of concepts in PBN. We hope this study shows that PBN, stimulated by and adapted from Dirac notation in Quantum Mechanics (QM), may have the potential to be a useful tool in probability modeling, at least for those who are already familiar with Dirac notation in QM.Comment: 37 pages, 5 figure