Editing Frequentist probability (section)

==Alternative views==
{{Main|Probability interpretations}}
[[Probability theory]] is a branch of mathematics. While its roots reach centuries into the past, it reached maturity with the axioms of [[Andrey Kolmogorov]] in 1933. The theory focuses on the valid operations on probability values rather than on the initial assignment of values; the mathematics is largely independent of any interpretation of probability.

Applications and interpretations of [[probability]] are considered by philosophy, the sciences and statistics. All are interested in the extraction of knowledge from observations—[[inductive reasoning]]. There are a variety of competing interpretations;<ref name=SEPIP>
{{cite encyclopedia
 |last = Hájek  |first = Alan
 |date = 21 October 2002
 |title = Interpretations of probability
 |editor-first = Edward N. |editor-last = Zalta
 |encyclopedia = The Stanford Encyclopedia of Philosophy
 |url = http://plato.stanford.edu/archives/win2012/entries/probability-interpret/
 |via = plato.stanford.edu
}}
</ref>
All have problems. The frequentist interpretation does resolve difficulties with the classical interpretation, such as any problem where the natural symmetry of outcomes is not known. It does not address other issues, such as the [[dutch book]].

* [[Classical definition of probability|Classical probability]] assigns probabilities based on physical idealized symmetry (dice, coins, cards). The classical definition is at risk of circularity: Probabilities are defined by assuming equality of probabilities.<ref name=Ash>
{{cite book
 |last = Ash  |first = Robert B.
 |year = 1970
 |title = Basic Probability Theory
 |publisher = Wiley
 |location = New York, NY
 |pages = 1–2
}}
</ref> In the absence of symmetry the utility of the definition is limited.

* [[Bayesian probability|Subjective (Bayesian) probability]] (a family of competing interpretations) considers degrees of belief: All practical "subjective" probability interpretations are so constrained to rationality as to avoid most subjectivity. Real subjectivity is repellent to some definitions of science which strive for results independent of the observer and analyst.{{citation needed|date=October 2019}} Other applications of Bayesianism in science (e.g. logical Bayesianism) embrace the inherent subjectivity of many scientific studies and objects and use Bayesian reasoning to place boundaries and context on the influence of [[Subjectivity#Sociology|subjectivities]] on all analysis.<ref>
{{cite journal
 |last1 = Fairfield  |first1 = Tasha
 |last2 = Charman    |first2 = Andrew E.
 |date = 15 May 2017
 |title = Explicit Bayesian analysis for process tracing: Guidelines, opportunities, and caveats
 |journal = [[Political Analysis (journal)|Political Analysis]]
 |volume = 25  |issue = 3  |pages = 363–380
 |doi = 10.1017/pan.2017.14  |s2cid = 8862619
 |url = http://eprints.lse.ac.uk/69203/
}}
</ref>  The historical roots of this concept extended to such non-numeric applications as legal evidence.

* [[Propensity probability]] views probability as a causative phenomenon rather than a purely descriptive or subjective one.<ref name=SEPIP/>