Beta-Binomial Bayesian Posterior and Credible Intervals

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Bayesian Estimation.

Your Task
A coin with unknown bias $\theta$ is tossed $n$ times, yielding $k$ heads. Under a uniform prior $\theta \sim \text{Beta}(1, 1)$, find the exact equal-tailed $95\%$ Bayesian credible interval for $\theta$. Compare the Bayesian credible interval with the frequentist Wald confidence interval $\hat{\theta} \pm 1.96 \sqrt{\frac{\hat{\theta}(1-\hat{\theta})}{n}}$, especially for small $k$.

Constraints

• All trials are mutually independent and identically distributed.
• Probabilities lie strictly within $(0, 1)$ unless degenerate.