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Derive MLE and Bayesian posterior for Bernoulli

OpenAIMachine Learning Engineer · Onsite
MediumMachine Learning EngineerOnsiteStatistics & Math

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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.

Given n Bernoulli trials with k successes: (a) derive the maximum likelihood estimator (MLE) for the success probability p and its asymptotic variance; (b) with a Beta(alpha, beta) prior, derive the posterior for p and the posterior predictive probability of success for the next trial; (c) compute a 95% confidence interval for p via normal approximation and a 95% credible interval from the posterior; (d) explain when each interval is reliable and how sample size affects inference.