{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:02:01Z","timestamp":1760144521725,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,4,28]],"date-time":"2024-04-28T00:00:00Z","timestamp":1714262400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>With this follow-up paper, we continue developing a mathematical framework based on information geometry for representing physical objects. The long-term goal is to lay down informational foundations for physics, especially quantum physics. We assume that we can now model information sources as univariate normal probability distributions N (\u03bc, \u03c30), as before, but with a constant \u03c30 not necessarily equal to 1. Then, we also relaxed the independence condition when modeling m sources of information. Now, we model m sources with a multivariate normal probability distribution Nm(\u03bc,\u03a30) with a constant variance\u2013covariance matrix \u03a30 not necessarily diagonal, i.e., with covariance values different to 0, which leads to the concept of modes rather than sources. Invoking Schr\u00f6dinger\u2019s equation, we can still break the information into m quantum harmonic oscillators, one for each mode, and with energy levels independent of the values of \u03c30, altogether leading to the concept of \u201cintrinsic\u201d. Similarly, as in our previous work with the estimator\u2019s variance, we found that the expectation of the quadratic Mahalanobis distance to the sample mean equals the energy levels of the quantum harmonic oscillator, being the minimum quadratic Mahalanobis distance at the minimum energy level of the oscillator and reaching the \u201cintrinsic\u201d Cram\u00e9r\u2013Rao lower bound at the lowest energy level. Also, we demonstrate that the global probability density function of the collective mode of a set of m quantum harmonic oscillators at the lowest energy level still equals the posterior probability distribution calculated using Bayes\u2019 theorem from the sources of information for all data values, taking as a prior the Riemannian volume of the informative metric. While these new assumptions certainly add complexity to the mathematical framework, the results proven are invariant under transformations, leading to the concept of \u201cintrinsic\u201d information-theoretic models, which are essential for developing physics.<\/jats:p>","DOI":"10.3390\/e26050370","type":"journal-article","created":{"date-parts":[[2024,4,29]],"date-time":"2024-04-29T10:33:36Z","timestamp":1714386816000},"page":"370","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Intrinsic Information-Theoretic Models"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4623-5935","authenticated-orcid":false,"given":"D.","family":"Bernal-Casas","sequence":"first","affiliation":[{"name":"Department of Genetics, Microbiology and Statistics, Faculty of Biology, Universitat de Barcelona, 08028 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9643-4406","authenticated-orcid":false,"given":"J. M.","family":"Oller","sequence":"additional","affiliation":[{"name":"Department of Genetics, Microbiology and Statistics, Faculty of Biology, Universitat de Barcelona, 08028 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bernal-Casas, D., and Oller, J.M. (2023). Information-Theoretic Models for Physical Observables. Entropy, 25.","DOI":"10.3390\/e25101448"},{"key":"ref_2","first-page":"309","article-title":"On the mathematical foundations of theoretical statistics","volume":"222","author":"Fisher","year":"1922","journal-title":"Philos. Trans. R. Soc. Lond. Ser. Contain. Pap. Math. Phys. Character"},{"key":"ref_3","unstructured":"Riemann, B. (2023, July 15). \u00dcber die Hypothesen, Welche der Geometrie zu Grunde Liegen. (Mitgetheilt durch R. Dedekind). 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Eigenvalues in Riemannian Geometry, Elsevier."},{"key":"ref_13","first-page":"370","article-title":"An essay towards solving a problem in the doctrine of chances","volume":"53","author":"Bayes","year":"1763","journal-title":"Phil. Trans. R. Soc. Lond."},{"key":"ref_14","first-page":"453","article-title":"An invariant form for the prior probability in estimation problems","volume":"186","author":"Jeffreys","year":"1946","journal-title":"Proc. R. Soc. Lond. Ser. Math. Phys. Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/5\/370\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:35:07Z","timestamp":1760106907000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/5\/370"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,28]]},"references-count":14,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["e26050370"],"URL":"https:\/\/doi.org\/10.3390\/e26050370","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,4,28]]}}}