= Jeffreys prior for a Bernoulli parameter
{c}
{title2=$\pi_J(\theta)=1/(\pi\sqrt{\theta(1-\theta)})$}
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> $\operatorname{Beta}(1/2,1/2)$, 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> $\operatorname{Beta}(s+1/2,n-s+1/2)$ 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.
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