Problem 3.8(b) Explanation

Conjugate Priors

The resulting posterior distribution has the form of a Beta distribution, denoted as $\text{Beta}(\alpha, \beta)$, where the PDF is proportional to $\pi^{\alpha-1}(1-\pi)^{\beta-1}$. Matching our result $p(\pi|\mathcal{D}) \propto \pi^s (1-\pi)^{n-s}$:

• $\alpha - 1 = s \implies \alpha = s + 1$
• $\beta - 1 = n - s \implies \beta = n - s + 1$

The fact that the posterior distribution is in the same family (Beta distribution) as the prior (Uniform distribution is actually $\text{Beta}(1,1)$) means the Beta distribution is the conjugate prior for the Bernoulli/Binomial likelihood. This property makes Bayesian updates analytically tractable.

Normalizing Constant

The term $\frac{(n+1)!}{s!(n-s)!}$ acts purely as a normalizing constant to ensure the area under the curve equals 1. If we recognize this as a Beta distribution $\text{Beta}(s+1, n-s+1)$, the standard normalizing constant is $\frac{1}{B(s+1, n-s+1)} = \frac{\Gamma(n+2)}{\Gamma(s+1)\Gamma(n-s+1)}$, which simplifies using factorials ($\Gamma(n) = (n-1)!$) to exactly what we derived.

Interpretation of $n=1$ Plot

• Prior: We started with a flat line ($p(\pi)=1$), meaning we had no reason to believe any value of $\pi$ was more likely than another.
• Data: We observed one coin flip.
• Posterior ($s=1$): We saw a Head. Now, values of $\pi$ near 1 are more likely than values near 0. The probability density increases linearly. It doesn't rule out $\pi=0.1$ completely (it's just unlikely), but it strongly suggests $\pi$ is high.
• Posterior ($s=0$): We saw a Tail. The belief is flipped. Values near 0 are now more likely.