Question

Problem 3.8 Bayesian estimation for a Bernoulli distribution

In this problem we will consider Bayesian estimation and prediction for a Bernoulli r.v. Let $x$ be a r.v. with a Bernoulli distribution, $p(x|\pi) = \pi^x (1 - \pi)^{1-x},$ where $\pi = P(x = 1)$ is the parameter.

(a) Let $\mathcal{D} = \{x_1, \cdots, x_n\}$ be a set of samples, show that $p(\mathcal{D}|\pi) = \pi^s (1 - \pi)^{n-s},$ where $s = \sum_{i=1}^n x_i$ is the sum of the samples (sufficient statistic).