page_title: Beta distribution
section_title: Haldane's prior probability (Beta(0,0))
hierarchical_section_title: Beta distribution / Statistical Inference / Bayesian inference / Haldane's prior probability (Beta(0,0))
caption_attribution_description: English: Beta distribution for alpha and beta approaching zero - J. Rodal
context_page_description: In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.
  The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines.
  In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. The beta distribution is a suitable model for the random behavior of percentages and proportions.
  The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution.
context_section_description: The Beta(0,0) distribution was proposed by J.B.S. Haldane, who suggested that the prior probability representing complete uncertainty should be proportional to p⁻¹(1−p)⁻¹. The function p⁻¹(1−p)⁻¹ 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⁻¹(1−p)⁻¹ 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 points out that on the log-odds scale, (the logit transformation ln(p/1−p)), the Haldane prior is the uniformly flat prior. The fact that a uniform prior probability on the logit transformed variable ln(p/1−p) (with domain (-∞, ∞)) is equivalent to the Haldane prior on the domain [0, 1] was pointed out by Harold Jeffreys in the first edition (1939) of his book Theory of Probability ( p. 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 dx/(x(1−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.
caption: B
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      {\displaystyle Beta(0,0)}
    
  : 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.
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