= Concentration of a beta prior near zero
{title2=$\mathbb P(\Theta>t)=(1-t)^\beta$}
For $\Theta\sim\operatorname{Beta}(1,\beta)$, direct <integration> of its <probability density function> gives the displayed tail for $0<t<1$. Increasing $\beta$ strictly decreases every such tail <probability>, and $\Theta$ converges in <probability> to zero as $\beta\to\infty$. Its <mean> is $1/(\beta+1)$ and its <variance> is $\beta/[(\beta+1)^2(\beta+2)]$; both decrease for $\beta>1$. For the <variance>, logarithmic <differentiation> gives $1/\beta-2/(\beta+1)-1/(\beta+2)<0$ in that range.
After $S$ successes in $N$ conditionally independent <Bernoulli trials>, <Beta-binomial conjugacy> gives <posterior mean> $(S+1)/(N+\beta+1)$. Its <derivative> with respect to $\beta$ is negative, while the weight $(\beta+1)/(N+\beta+1)$ assigned to the <prior mean> is increasing for $N>0$.
Back to article page
