Editing Bayesian probability (section)

==Personal probabilities and objective methods for constructing priors{{Anchor|subjective}}==
Following the work on [[expected utility]] [[optimal decision|theory]] of [[Frank P. Ramsey|Ramsey]] and [[John von Neumann|von Neumann]], decision-theorists have accounted for [[optimal decision|rational behavior]] using a probability distribution for the [[Agent-based model|agent]]. [[Johann Pfanzagl]] completed the ''[[Theory of Games and Economic Behavior]]'' by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and [[Oskar Morgenstern]]: their original theory supposed that all the agents had the same probability distribution, as a convenience.<ref>Pfanzagl (1967, 1968)</ref> Pfanzagl's axiomatization was endorsed by Oskar Morgenstern: "Von Neumann and I have anticipated ... [the question whether probabilities] might, perhaps more typically, be subjective and have stated specifically that in the latter case axioms could be found from which could derive the desired numerical utility together with a number for the probabilities (cf. p. 19 of The Theory of Games and Economic Behavior). We did not carry this out; it was demonstrated by Pfanzagl ... with all the necessary rigor".<ref>Morgenstern (1976, page 65)</ref>

Ramsey and [[Leonard Jimmie Savage|Savage]] noted that the individual agent's probability distribution could be objectively studied in experiments. Procedures for [[statistical hypothesis testing|testing hypotheses]] about probabilities (using finite samples) are due to [[Frank P. Ramsey|Ramsey]] (1931) and [[Bruno de Finetti|de Finetti]] (1931, 1937, 1964, 1970). Both [[Bruno de Finetti]]<ref>{{Cite journal |last=Galavotti |first=Maria Carla|author-link= Maria Carla Galavotti |date=1989-01-01 |title=Anti-Realism in the Philosophy of Probability: Bruno de Finetti's Subjectivism |journal=Erkenntnis |volume=31 |issue=2/3 |pages=239–261 |doi=10.1007/bf01236565 |jstor=20012239 |s2cid=170802937 |df=dmy-all}}</ref><ref name=":0">{{Cite journal |last=Galavotti |first=Maria Carla|author-link= Maria Carla Galavotti |date=1991-12-01 |title=The notion of subjective probability in the work of Ramsey and de Finetti |journal=Theoria |language=en |volume=57 |issue=3 |pages=239–259 |doi=10.1111/j.1755-2567.1991.tb00839.x |issn=1755-2567 |df=dmy-all}}</ref> and [[Frank P. Ramsey]]<ref name=":0" /><ref name=":1">{{Cite book |title=Frank Ramsey: Truth and Success |last1=Dokic |first1=Jérôme |last2=Engel |first2=Pascal |publisher=Routledge |year=2003 |isbn=9781134445936}}</ref> acknowledge their debts to [[pragmatic philosophy]], particularly (for Ramsey) to [[Charles Sanders Peirce|Charles S. Peirce]].<ref name=":0" /><ref name=":1" />

The "Ramsey test" for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century.<ref>Davidson et al. (1957)</ref>
This work demonstrates that Bayesian-probability propositions can be [[Falsifiability|falsified]], and so meet an empirical criterion of [[Charles Sanders Peirce|Charles S. Peirce]], whose work inspired Ramsey. (This [[falsifiability]]-criterion was popularized by [[Karl Popper]].<ref>{{cite encyclopedia |url=http://plato.stanford.edu/entries/popper/#ProDem |article=Karl Popper |title=Stanford Encyclopedia of Philosophy|first = Stephen |last =Thornton |date = 7 August 2018|publisher=Metaphysics Research Lab, Stanford University }}</ref><ref>{{cite book |author=Popper, Karl |year=2002 |url=https://books.google.com/books?id=T76Zd20IYlgC&q=logic+of+scientific+discovery |via=Google Books |title=The Logic of Scientific Discovery |edition=2nd |publisher=Routledge |isbn=0-415-27843-0 |page=57 |orig-year=1959 |language=en}} (translation of 1935 original, in German).</ref>)

Modern work on the experimental evaluation of personal probabilities uses the randomization, [[double blind|blinding]], and Boolean-decision procedures of the Peirce-Jastrow experiment.<ref>Peirce & Jastrow (1885)
</ref> Since individuals act according to different probability judgments, these agents' probabilities are "personal" (but amenable to objective study).

Personal probabilities are problematic for science and for some applications where decision-makers lack the knowledge or time to specify an informed probability-distribution (on which they are prepared to act). To meet the needs of science and of human limitations, Bayesian statisticians have developed "objective" methods for specifying prior probabilities.

Indeed, some Bayesians have argued the prior state of knowledge defines ''the'' (unique) prior probability-distribution for "regular" statistical problems; cf. [[well-posed problem]]s. Finding the right method for constructing such "objective" priors (for appropriate classes of regular problems) has been the quest of statistical theorists from Laplace to [[John Maynard Keynes]], [[Harold Jeffreys]], and [[Edwin Thompson Jaynes]]. These theorists and their successors have suggested several methods for constructing "objective" priors (Unfortunately, it is not always clear how to assess the relative "objectivity" of the priors proposed under these methods):
* [[Principle of maximum entropy|Maximum entropy]]
* [[Haar measure|Transformation group analysis]]
* [[José-Miguel Bernardo|Reference analysis]]

Each of these methods contributes useful priors for "regular" one-parameter problems, and each prior can handle some challenging [[statistical model]]s (with "irregularity" or several parameters). Each of these methods has been useful in Bayesian practice. Indeed, methods for constructing "objective" (alternatively, "default" or "ignorance") priors have been developed by avowed subjective (or "personal") Bayesians like [[James Berger (statistician)|James Berger]] ([[Duke University]]) and [[José-Miguel Bernardo]] ([[University of Valencia|Universitat de València]]), simply because such priors are needed for Bayesian practice, particularly in science.<ref name="refa">{{cite book |author=Bernardo, J. M. |year=2005 |url=http://www.uv.es/~bernardo/RefAna.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.uv.es/~bernardo/RefAna.pdf |archive-date=2022-10-09 |url-status=live |article=Reference Analysis |title=Handbook of Statistics |volume=25 |editor1-first=D.K. |editor1-last=Dey |editor2-link=C. R. Rao |editor2-first=C. R. |editor2-last=Rao |location=Amsterdam |publisher=Elsevier |pages=17–90}}</ref> The quest for "the universal method for constructing priors" continues to attract statistical theorists.<ref name="refa" />

Thus, the Bayesian statistician needs either to use informed priors (using relevant expertise or previous data) or to choose among the competing methods for constructing "objective" priors.