{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:23:06Z","timestamp":1772252586031,"version":"3.50.1"},"reference-count":70,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,14]],"date-time":"2022-10-14T00:00:00Z","timestamp":1665705600000},"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>Philosophers frequently define knowledge as justified, true belief. We built a mathematical framework that makes it possible to define learning (increasing number of true beliefs) and knowledge of an agent in precise ways, by phrasing belief in terms of epistemic probabilities, defined from Bayes\u2019 rule. The degree of true belief is quantified by means of active information I+: a comparison between the degree of belief of the agent and a completely ignorant person. Learning has occurred when either the agent\u2019s strength of belief in a true proposition has increased in comparison with the ignorant person (I+&gt;0), or the strength of belief in a false proposition has decreased (I+&lt;0). Knowledge additionally requires that learning occurs for the right reason, and in this context we introduce a framework of parallel worlds that correspond to parameters of a statistical model. This makes it possible to interpret learning as a hypothesis test for such a model, whereas knowledge acquisition additionally requires estimation of a true world parameter. Our framework of learning and knowledge acquisition is a hybrid between frequentism and Bayesianism. It can be generalized to a sequential setting, where information and data are updated over time. The theory is illustrated using examples of coin tossing, historical and future events, replication of studies, and causal inference. It can also be used to pinpoint shortcomings of machine learning, where typically learning rather than knowledge acquisition is in focus.<\/jats:p>","DOI":"10.3390\/e24101469","type":"journal-article","created":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T00:04:55Z","timestamp":1665965095000},"page":"1469","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Formal Framework for Knowledge Acquisition: Going beyond Machine Learning"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2767-8818","authenticated-orcid":false,"given":"Ola","family":"H\u00f6ssjer","sequence":"first","affiliation":[{"name":"Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6281-1720","authenticated-orcid":false,"given":"Daniel Andr\u00e9s","family":"D\u00edaz-Pach\u00f3n","sequence":"additional","affiliation":[{"name":"Division of Biostatistics, University of Miami, Miami, FL 33136, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6450-3200","authenticated-orcid":false,"given":"J. Sunil","family":"Rao","sequence":"additional","affiliation":[{"name":"Division of Biostatistics, University of Miami, Miami, FL 33136, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,14]]},"reference":[{"key":"ref_1","unstructured":"Piech, C., Bassen, J., Huang, J., Ganguli, S., Sahami, M., Guibas, L.J., and Sohl-Dickstein, J. (2015, January 7\u201312). Deep Knowledge Tracing. Proceedings of the Neural Information Processing Systems (NIPS) 2015, Montreal, QC, Canada."},{"key":"ref_2","unstructured":"Zalta, E.N. (2021). Knowledge How. The Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1007\/s10955-017-1892-x","article-title":"Phase transition for the Maki-Thompson rumour model on a small-world network","volume":"169","author":"Agliari","year":"2017","journal-title":"J. Stat. 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