Editing Beta distribution (section)

===={{Anchor|Haldane prior}}Haldane's prior probability (Beta(0,0))====
[[File:Beta distribution for alpha and beta approaching zero - J. Rodal.png|thumb|<math>Beta(0,0)</math>: The Haldane prior probability expressing total ignorance about prior information, where we are not even sure whether it is physically possible for an experiment to yield either a success or a failure. As α, β → 0, the beta distribution approaches a two-point [[Bernoulli distribution]] with all probability density concentrated at each end, at 0 and 1, and nothing in between. A coin-toss: one face of the coin being at 0 and the other face being at 1. ]]

The Beta(0,0) distribution was proposed by [[J.B.S. Haldane]],<ref>{{cite journal|last=Haldane |first=J. B. S.| authorlink1=J. B. S. Haldane |title=A note on inverse probability|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|year=1932|volume=28|issue=1|pages=55–61|doi=10.1017/s0305004100010495|bibcode=1932PCPS...28...55H|s2cid=122773707 }}</ref> who suggested that the prior probability representing complete uncertainty should be proportional to ''p''<sup>−1</sup>(1−''p'')<sup>−1</sup>. The function ''p''<sup>−1</sup>(1−''p'')<sup>−1</sup> can be viewed as the limit of the numerator of the beta distribution as both shape parameters approach zero: α, β → 0. The Beta function (in the denominator of the beta distribution) approaches infinity, for both parameters approaching zero, α, β → 0. Therefore, ''p''<sup>−1</sup>(1−''p'')<sup>−1</sup> divided by the Beta function approaches a 2-point [[Bernoulli distribution]] with equal probability 1/2 at each end, at 0 and 1, and nothing in between, as α, β → 0. A coin-toss: one face of the coin being at 0 and the other face being at 1.  The Haldane prior probability distribution Beta(0,0) is an "[[improper prior]]" because its integration (from 0 to 1) fails to strictly converge to 1 due to the singularities at each end. However, this is not an issue for computing posterior probabilities unless the sample size is very small.  Furthermore, Zellner<ref name=Zellner>{{cite book|last=Zellner |first=Arnold|title=An Introduction to Bayesian Inference in Econometrics|year=1971|publisher=Wiley-Interscience|isbn=978-0471169376}}</ref> points out that on the [[log-odds]] scale, (the [[logit]] transformation <math>\log(p/(1-p))</math>), the Haldane prior is the uniformly flat prior. The fact that a uniform prior probability on the [[logit]] transformed variable ln(''p''/1&nbsp;−&nbsp;''p'') (with domain (−∞, ∞)) is equivalent to the Haldane prior on the domain&nbsp;[0,&nbsp;1] was pointed out by [[Harold Jeffreys]] in the first edition (1939) of his book Theory of Probability (<ref name=Jeffreys/> p.&nbsp;123).  Jeffreys writes "Certainly if we take the Bayes–Laplace rule right up to the extremes we are led to results that do not correspond to anybody's way of thinking. The (Haldane) rule d''x''/(''x''(1&nbsp;−&nbsp;''x'')) goes too far the other way.  It would lead to the conclusion that if a sample is of one type with respect to some property there is a probability 1 that the whole population is of that type."  The fact that "uniform" depends on the parametrization, led Jeffreys to seek a form of prior that would be invariant under different parametrizations.