This article$^1$ on p.16 specifies Haldane's prior as:
$$p(theta) = frac{1}{θ(1−θ)}$$.
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
Best Answer
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.