Editing Beta distribution (section)

====Rule of succession====
{{Main|Rule of succession}}
A classic application of the beta distribution is the [[rule of succession]], introduced in the 18th century by [[Pierre-Simon Laplace]]<ref name=Laplace>{{cite book|last=Laplace|first=Pierre Simon, marquis de|title=A philosophical essay on probabilities|year=1902|publisher=New York : J. Wiley; London : Chapman & Hall|isbn=978-1-60206-328-0|url=https://archive.org/details/philosophicaless00lapliala}}</ref>  in the course of treating the [[sunrise problem]].  It states that, given ''s'' successes in ''n'' [[conditional independence|conditionally independent]] [[Bernoulli trial]]s with probability ''p,'' that the estimate of the expected value in the next trial is <math>\frac{s+1}{n+2}</math>.  This estimate is the expected value of the posterior distribution over ''p,'' namely Beta(''s''+1, ''n''−''s''+1), which is given by [[Bayes' rule]] if one assumes a uniform prior probability over ''p'' (i.e., Beta(1, 1)) and then observes that ''p'' generated ''s'' successes in ''n'' trials.  Laplace's rule of succession has been criticized by prominent scientists.  R. T. Cox described Laplace's application of the rule of succession to the [[sunrise problem]] (<ref name=CoxRT>{{cite book|last=Cox|first=Richard T.|title=Algebra of Probable Inference|year=1961|publisher=The Johns Hopkins University Press|isbn=978-0801869822}}</ref> p.&nbsp;89) as "a travesty of the proper use of the principle".  Keynes remarks  (<ref name=KeynesTreatise>{{cite book|last=Keynes|first=John Maynard|title=A Treatise on Probability: The Connection Between Philosophy and the History of Science|orig-year=1921|year=2010|publisher=Wildside Press|isbn=978-1434406965}}</ref> Ch.XXX, p.&nbsp;382)  "indeed this is so foolish a theorem that to entertain it is discreditable".  Karl Pearson<ref name=PearsonRuleSuccession>{{cite journal|last=Pearson|first=Karl|title=On the Influence of Past Experience on Future Expectation|journal=Philosophical Magazine|year=1907|volume=6|issue=13|pages=365–378}}</ref>  showed that the probability that the next (''n''&nbsp;+&nbsp;1) trials will be successes, after n successes in n trials, is only 50%, which has been considered too low by scientists like Jeffreys and unacceptable as a representation of the scientific process of experimentation to test a proposed scientific law.  As pointed out by Jeffreys (<ref name=Jeffreys/> p.&nbsp;128) (crediting [[C. D. Broad]]<ref name=BroadMind>{{cite journal|last=Broad|first=C. D.|title=On the relation between induction and probability|journal=MIND, A Quarterly Review of Psychology and Philosophy|date=October 1918|volume=27 (New Series)|issue=108|pages=389–404|jstor=2249035|doi=10.1093/mind/XXVII.4.389}}</ref> ) Laplace's rule of succession establishes a high probability of success ((n+1)/(n+2)) in the next trial, but only a moderate probability (50%) that a further sample (n+1) comparable in size will be equally successful.  As pointed out by Perks,<ref name=Perks>{{cite journal|last=Perks|first=Wilfred|title=Some observations on inverse probability including a new indifference rule|journal=Journal of the Institute of Actuaries|date=January 1947|volume=73|issue=2|pages=285–334|url=http://www.actuaries.org.uk/research-and-resources/documents/some-observations-inverse-probability-including-new-indifference-ru|doi=10.1017/S0020268100012270|access-date=2012-09-19|archive-date=2014-01-12|archive-url=https://web.archive.org/web/20140112111032/http://www.actuaries.org.uk/research-and-resources/documents/some-observations-inverse-probability-including-new-indifference-ru|url-status=dead}}</ref> "The rule of succession itself is hard to accept. It assigns a probability to the next trial which implies the assumption that the actual run observed is an average run and that we are always at the end of an average run. It would, one would think, be more reasonable to assume that we were in the middle of an average run. Clearly a higher value for both probabilities is necessary if they are to accord with reasonable belief." These problems with Laplace's rule of succession motivated Haldane, Perks, Jeffreys and others to search for other forms of prior probability (see the next {{section link||Bayesian inference}}).  According to Jaynes,<ref name=Jaynes/> the main problem with the rule of succession is that it is not valid when s=0 or s=n (see [[rule of succession]], for an analysis of its validity).