Haldane prior ($\pi (p)\propto [p(1-p){]}^{-1}$)

The Haldane prior has formal beta distribution parameters $(0,0)$. It is an improper prior, with infinite mass at both endpoints of $(0,1)$. For a binomial distribution likelihood with $s$ successes and $f$ failures, its posterior is $\mathrm{Beta}(s,f)$ only when $s,f>0$. A density flat in the log odds induces this kernel by the Jacobian determinant $d\log [p/(1-p)]/dp=1/[p(1-p)]$.