{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:59:15Z","timestamp":1760234355734,"version":"build-2065373602"},"reference-count":61,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,13]],"date-time":"2021-05-13T00:00:00Z","timestamp":1620864000000},"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>When humans infer underlying probabilities from stochastic observations, they exhibit biases and variability that cannot be explained on the basis of sound, Bayesian manipulations of probability. This is especially salient when beliefs are updated as a function of sequential observations. We introduce a theoretical framework in which biases and variability emerge from a trade-off between Bayesian inference and the cognitive cost of carrying out probabilistic computations. We consider two forms of the cost: a precision cost and an unpredictability cost; these penalize beliefs that are less entropic and less deterministic, respectively. We apply our framework to the case of a Bernoulli variable: the bias of a coin is inferred from a sequence of coin flips. Theoretical predictions are qualitatively different depending on the form of the cost. A precision cost induces overestimation of small probabilities, on average, and a limited memory of past observations, and, consequently, a fluctuating bias. An unpredictability cost induces underestimation of small probabilities and a fixed bias that remains appreciable even for nearly unbiased observations. The case of a fair (equiprobable) coin, however, is singular, with non-trivial and slow fluctuations in the inferred bias. The proposed framework of costly Bayesian inference illustrates the richness of a \u2018resource-rational\u2019 (or \u2018bounded-rational\u2019) picture of seemingly irrational human cognition.<\/jats:p>","DOI":"10.3390\/e23050603","type":"journal-article","created":{"date-parts":[[2021,5,13]],"date-time":"2021-05-13T11:10:06Z","timestamp":1620904206000},"page":"603","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Biases and Variability from Costly Bayesian Inference"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6710-1488","authenticated-orcid":false,"given":"Arthur","family":"Prat-Carrabin","sequence":"first","affiliation":[{"name":"Department of Economics, Columbia University, New York, NY 10027, USA"},{"name":"Laboratoire de Physique de l\u2019\u00c9cole Normale Sup\u00e9rieure, Universit\u00e9 Paris Sciences &amp; Lettres, Centre National de la Recherche Scientifique, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6992-678X","authenticated-orcid":false,"given":"Florent","family":"Meyniel","sequence":"additional","affiliation":[{"name":"Cognitive Neuroimaging Unit, Institut National de la Sant\u00e9 et de la Recherche M\u00e9dicale, Commissariat \u00e0 l\u2019Energie Atomique et aux Energies Alternatives, Universit\u00e9 Paris-Saclay, NeuroSpin Center, 91191 Gif-sur-Yvette, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5661-4349","authenticated-orcid":false,"given":"Misha","family":"Tsodyks","sequence":"additional","affiliation":[{"name":"Department of Neurobiology, Weizmann Institute of Science, Rehovot 76000, Israel"},{"name":"The Simons Center for Systems Biology, Institute for Advanced Study, Princeton, NJ 08540, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8487-4105","authenticated-orcid":false,"given":"Rava","family":"Azeredo da Silveira","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique de l\u2019\u00c9cole Normale Sup\u00e9rieure, Universit\u00e9 Paris Sciences &amp; Lettres, Centre National de la Recherche Scientifique, 75005 Paris, France"},{"name":"Department of Neurobiology, Weizmann Institute of Science, Rehovot 76000, Israel"},{"name":"Institute of Molecular and Clinical Ophthalmology Basel, 4056 Basel, Switzerland"},{"name":"Faculty of Science, University of Basel, 4001 Basel, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,13]]},"reference":[{"unstructured":"Wikipedia (2020, September 19). 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