= Solution
{c}
The <prior mean> $1/(\beta+1)$ decreases as $\beta$ increases. More strongly, its <cumulative distribution function> and upper-tail <probability> are
$$
F_\beta(t)=1-(1-t)^\beta,\qquad
\mathbb P_\beta(\theta>t)=(1-t)^\beta,\qquad 0<t<1.
$$
For every fixed $t$, that upper-tail <probability> decreases with $\beta$. The statistician increasingly favors smaller values of $\theta$; as $\beta\to\infty$, the <prior distribution> concentrates near zero. For $\beta>1$, its <variance> $\beta/[(\beta+1)^2(\beta+2)]$ also decreases. This is <concentration of a beta prior near zero>.
With the data fixed, the <derivatives> are
$$
\boxed{\frac{d\lambda}{d\beta}=\frac{N}{(N+\beta+1)^2}>0},\qquad
\boxed{\frac{d\widetilde\theta_1}{d\beta}
=-\frac{S+1}{(N+\beta+1)^2}<0}.
$$
Thus \b[the prior weight increases and the posterior mean decreases]. Their limits are $\lambda\to1$ and $\widetilde\theta_1\to0$. Both the changing <prior mean> and the increasing weight contribute to the change in the <posterior mean>.
Back to article page
