Article
Version 1
This version is not peer-reviewed
Probability via Expectation Measures
Version 1
: Received: 23 October 2024 / Approved: 24 October 2024 / Online: 24 October 2024 (10:27:22 CEST)
How to cite: Harremoës, P. Probability via Expectation Measures. Preprints 2024, 2024101897. https://doi.org/10.20944/preprints202410.1897.v1 Harremoës, P. Probability via Expectation Measures. Preprints 2024, 2024101897. https://doi.org/10.20944/preprints202410.1897.v1
Abstract
Since the seminal work of Kolmogorov, probability theory has been based on measure theory, where the central components are so-called probability measures, defined as measures with total mass equal to 1. In Kolmogorov’s theory, a probability measure is used to model an experiment with a single outcome that will belong to exactly one out of several disjoint sets. In this paper, we present a different basic model where an experiment results in a multiset, i.e. for each of the disjoint sets we get the number of observations in the set. This new framework is consistent with Kolmogorov’s theory, but the theory focuses on expected values rather than probabilities. We present examples from testing Goodness-of-Fit, Bayesian statistics, and quantum theory, where the shifted focus gives new insight or better performance. We also provide several new theorems that address some problems related to the change in focus.
Keywords
Category; double slit experiment; expectation measure; extended probabilistic power domain; expected value; Gaussian approximation; information divergence; monad; point process; Poisson distribution; Poisson point process; quantum information theory; thinning; valuation
Subject
Computer Science and Mathematics, Probability and Statistics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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