# Solved – Haldane’s prior Beta(0,0) – Part 1

\$\$p(theta) = frac{1}{θ(1−θ)}\$\$.

However, other\$^2\$ source on p.6 specifies Haldane's prior as proportional to \$frac{1}{θ(1−θ)}\$, i.e.
\$\$p(theta) propto frac{1}{θ(1−θ)}\$\$.

Could anyone clarify which expression is the accurate one.

1. Approximation of improper priors

2. Bayesian Analysis of Some Common Distributions

Contents

Haldane prior is beta distribution with parameters \$alpha = beta = 0\$. So it is

\$\$ f(p) = frac{p^{alpha-1} (1-p)^{beta-1}}{B(alpha, beta)} = frac{p^{-1}(1-p)^{-1}}{B(0, 0)} \$\$

where \$B(0, 0)\$ is the normalizing constant that is infinite as described in Wikipedia:

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: \$alpha, beta to 0\$. The Beta function (in the denominator of the beta distribution) approaches infinity, for both parameters approaching zero, \$alpha, beta to 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 Dirac delta function end, at \$0\$ and \$1\$, and nothing in between, as \$alpha, beta to 0\$.

So Haldane prior is not a proper distribution. It is an abstract idea of what would be the beta distribution be if it had \$alpha = beta = 0\$ parameters. As a distribution, it is rather not usable, yet it can be used as an "uninformative" prior for binomial distribution. It is often described in it's approximate form \$f(p) propto p^{-1}(1-p)^{-1}\$, since the normalizing constant is meaningless.

Rate this post