 Solution

The uniform prior and likelihood give the posterior densitya Beta distribution with parameters $K+1,n-K+1$. Under squared-error loss for $\varphi$, the posterior expected loss is $\mathbb{E}[{\varphi }^2\mid K]-2a\mathbb{E}[\varphi \mid K]+a^2$, minimized uniquely at the posterior mean of $\varphi$, not at the product of the two posterior mean factors.

Using the supplied beta integral,Thus the Bayes estimate isThis remains positive even on the endpoint samples, unlike the extended MLE; it integrates over the posterior uncertainty in $\theta$.