Jeffreys prior for a Bernoulli parameter (${\pi }_J(\theta )=1/(\pi \sqrt{\theta (1-\theta )})$)

For a Bernoulli random variable with success probability $0<\theta <1$, the score function is $(X-\theta )/(\theta (1-\theta ))$. The expected value of its square is $I(\theta )=1/(\theta (1-\theta ))$, since the variance of $X$ is $\theta (1-\theta )$. The Jeffreys prior is therefore the proper beta distribution $\mathrm{Beta}(1/2,1/2)$, also called the arcsine distribution. The Euler beta function has $B(1/2,1/2)=\pi$, giving the stated normalized prior density.

After $s$ successes in $n$ independent Bernoulli trials, Beta-binomial conjugacy gives posterior distribution $\mathrm{Beta}(s+1/2,n-s+1/2)$ and posterior predictive probability $(s+1/2)/(n+1)$ of one further success. Although its prior density diverges at both boundaries, its integral is finite.