Related papers: Probability Bracket Notation: the Unified Expressi…
We extend Probability Bracket Notation (PBN), inspired by the Dirac notation in quantum mechanics, to multivariable probability systems and static Bayesian networks (BNs). By defining probability distributions and conditional expectations…
Following the Dirac Notation in Quantum Mechanics (QM), we propose the 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…
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…
Inspired by the Dirac vector probability notation (VPN), we propose the Probability Bracket Notation (PBN), a new set of symbols defined similarly (but not identically) as in the VPN. Applying the PBN to fundamental definitions and theorems…
Following the Dirac vector bracket notation (VBN), we proposed the probability bracket notation (PBN) in our previous paper. We mentioned that under the special Wick rotation (imaginary time), a stationary Schrodinger equation in the…
In this work, we advance the development of the Probability Bracket Notation (PBN), a formalism inspired by Dirac's notation in quantum mechanics, to provide a unified framework for probability modeling. We demonstrate that under a Special…
This paper is devoted to clarification of the notion of entanglement through decoupling it from the tensor product structure and treating as a constraint posed by probabilistic dependence of quantum observable A and B. In our framework, it…
Quantum probabilities are defined for several important physical cases characterizing measurements with multimode quantum systems. These are the probabilities for operationally testable measurements, for operationally uncertain…
Based on a recent proof of free choices in linking equations to the experiments they describe, I clarify relations among some purely mathematical entities featured in quantum mechanics (probabilities, density operators, partial traces, and…
Our aim is to make a step towards clarification of foundations for the notion of entanglement (both physical and mathematical) by representing it in the conditional probability framework. In Schr\"odinger's words, this is entanglement of…
In this paper, we introduce and develop the concept of conditional quantization for Borel probability measures on $\mathbb{R}^k,$ considering both constrained and unconstrained frameworks. For each setting, we define the associated…
Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the support are…
Observables and instruments have played significant roles in recent studies on the foundations of quantum mechanics. Sequential products of effects and conditioned observables have also been introduced. After an introduction in Section~1,…
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…
We introduce a contextual quantum system comprising mutually complementary observables organized into two or more collections of pseudocontexts with the same probability sums of outcomes. These pseudocontexts constitute non-orthogonal bases…
We explore further the suggestion to describe a pre- and post-selected system by a two-state, which is determined by two conditions. Starting with a formal definition of a two-state Hilbert space and basic operations, we systematically…
Probabilistic conditioning is concerned with the identification of a distribution of a random variable $X$ given a random variable $Y$. It is a cornerstone of scientific and engineering applications where modeling uncertainty is key. This…
We propose that a quantum particle in a potential in one space dimension can be described by a probabilistic cellular automaton. While the simple updating rule of the automaton is deterministic, the probabilistic description is introduced…
In the Contextuality-by-Default theory random variables representing measurement outcomes are labeled contextually, i.e., not only by what they measure but also under what conditions (in what contexts) the measurements are made, including…
We show that including both the system and the apparatus in the quantum description of the measurement process, and using the concept of conditional probabilities, it is possible to deduce the statistical operator of the system after a…