Solution
= Solution
The <binomial likelihood> is $p(x\mid\theta)=\binom mx\theta^x(1-\theta)^{m-x}$. Multiplication by the <Beta distribution> <prior density> and <Bayes' theorem> give the <posterior density> kernel $\theta^{a+x-1}(1-\theta)^{b+m-x-1}$. Normalizing by the <beta function> yields <Beta-binomial conjugacy>:
$$
\boxed{\theta\mid x\sim\operatorname{Beta}(a+x,b+m-x).}
$$
This is proper for $a,b>0$ and $0\leq x\leq m$, including counts at either boundary.