Final Exam Sheet

The Frequentist Perspective:

• Probability represents the long-run relative frequency of random events.
• Parameters are considered (often unknown) fixed constants.

The Bayesian Perspective:

• Probability represents one’s subjective belief about random events.
• Parameters are considered random variables.

\[ P(\theta \mid y) = \frac{P(y \mid \theta)\,P(\theta)} {P(y)} \]

Beta-Binomial Model

If

\[ Y \mid \theta \sim \text{Bin}(n,\theta), \qquad \theta \sim \text{Beta}(\alpha,\beta) \]

then the posterior distribution is

\[ \theta \mid y \sim \text{Beta}(\alpha+y,\;\beta+n-y) \]

Posterior mean:

\[ E(\theta \mid y) = \frac{\alpha+y}{\alpha+\beta+n} \]