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Question

Problem 3.8(c)​

Given the posterior in (3.33), show that the predictive distribution is

p(x∣D)=(s+1n+2)x(1−s+1n+2)1−x.(3.34)p(x|\mathcal{D}) = \left(\frac{s+1}{n+2}\right)^x \left(1-\frac{s+1}{n+2}\right)^{1-x}. \quad (3.34)

What is the effective Bayesian estimate of π\pi? What is the intuitive explanation in terms of "virtual" samples added to an equivalent MLE estimate?