Maximum A Posteriori (MAP) Estimation for Bernoulli Parameter
Statistical Inference DS practice problem on Onlearn.
Difficulty: medium.
Topics: Maximum A Posteriori (MAP) Estimation for Bernoulli Parameter, Beta Distribution, Bernoulli Trial, Log-Likelihood, Hyperparameter Tuning, Maximum A Posteriori Estimation, Probability Theory, Statistical Inference, Optimization Theory, Bayesian Statistics, Information Theory, Point Estimation, Conjugate Priors, Likelihood Functions, Regularization Techniques, Probability Distributions.
Implement a function that computes the Maximum A Posteriori (MAP) estimate for the probability parameter of a Bernoulli distribution given binary observations and a Beta prior. In Bayesian inference, the MAP estimate is the value that maximizes the posterior distribution. Given: A list of binary observations (0s and 1s) representing outcomes from a Bernoulli process A Beta prior distribution with parameters alpha and beta representing prior beliefs about the success probability Your function should compute the MAP estimate of the success probability theta. The Beta distribution is the conjugate prior for the Bernoulli likelihood, which means the posterior distribution is also a Beta distribution. The MAP estimate corresponds to the mode (peak) of this posterior distribution. Your implementation should: 1. Compute the posterior distribution parameters 2. Find the mode of the posterior distribution 3. Handle edge cases where the mode lies at the boundaries (0 or 1) 4. Return the result rounded to 4 decimal places Note: Assume alpha = 1 and beta = 1 for the prior parameters.