Find $\mathbb{E}(\pi|S_n=k)$ knowing random variable $\pi$ is uniformly distributed over $(0,1)$ and $X_1,\ldots,X_n$ are conditionaly independent.

Question

Find $\mathbb{E}(\pi|S_n=k)$ knowing random variable $\pi$ is uniformly distributed over $(0,1)$, for every $1\leqslant i \leqslant n$ random variables $X_1,\ldots,X_n$ satisfy: $$\mathbb{P}(X_i=1|\pi=p) = p, \mathbb{P}(X_i=0|\pi=p) = 1-p $$ also they are conditionaly independent; $$ \mathbb{P} (X_1=\varepsilon_1,\ldots,X_n=\varepsilon_n | \pi)=\mathbb{P} (X_1=\varepsilon_1|\pi)\cdot\ldots\cdot\mathbb{P} (X_n=\varepsilon_n|\pi) $$ It is clear to me that since $X_1,\ldots,X_n$ are conditionaly independent it follows that $\mathbb{E}(S_n|\pi) = n\pi$ but I don't know how it could be used in this problem.

I suppose the answer is $\mathbb{E}(\pi|S_n) = \frac{S_n}{n}$ but how could it be proved?

Any hint would be appreciated.

Answer 1

Note $S_n|\pi\sim \text{Binomial}(n,\pi)$ and from Bayes' rule $$f_{\pi|S_n}(p|k)=\frac{f_{S_n|\pi}(k|p)f_{\pi}(p)}{\int_0^1f_{S_n|\pi}(k|p)f_{\pi}(p)\mathrm{d}p}=\frac{p^k(1-p)^{n-k}}{\int_0^1p^k(1-p)^{n-k}\mathrm{d}p}$$ Evidently $\pi|S_n \sim \text{Beta}(S_n+1,n-S_n+1)$ whose expected value is $$\mathbb{E}(\pi | S_n)=\frac{S_n+1}{(S_n+1)+(n-S_n+1)}=\frac{S_n+1}{n+2}$$