Derive MLE and Bayesian posterior for Bernoulli
Overview
This question evaluates parameter estimation and inferential reasoning for Bernoulli/binomial data, including maximum likelihood estimation and its asymptotic variance, Bayesian updating with a Beta prior and the resulting posterior and posterior predictive probability, as well as interval estimation; it is classified in the Statistics & Math domain. It is commonly asked to assess understanding of frequentist versus Bayesian inference, the behavior and reliability of asymptotic approximations and credible intervals, and the practical implications for predictive probability and uncertainty quantification, testing both conceptual understanding and practical application.
Bernoulli/Binomial Inference Task
You observe n independent Bernoulli trials with unknown success probability p, and you record k successes (so K ~ Binomial(n, p)).
Tasks
(a) Derive the maximum likelihood estimator (MLE) of p and its asymptotic variance.
(b) Assume a Beta(alpha, beta) prior on p. Derive the posterior distribution of p and the posterior predictive probability that the next trial is a success.
(c) Compute a 95% confidence interval (CI) for p using the normal approximation, and a 95% credible interval from the posterior in (b).
(d) Explain when each interval (Wald CI vs. Bayesian credible interval) is reliable and how sample size affects the inference.
Solution
Login required