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,

$$p(x|\pi) = \pi^x(1-\pi)^{1-x}, \quad (3.30)$$

where $\pi = P(x=1)$ is the parameter.

(a) Let $\mathcal{D} = \{x_1, \cdots, x_n\}$ be a set of samples, show that

$$p(\mathcal{D}|\pi) = \pi^s(1-\pi)^{n-s}, \quad (3.31)$$

where $s = \sum_{i=1}^n x_i$ is the sum of the samples (sufficient statistic).