Frequentist Versus Bayesian Parameter Estimation of The Beta Distribution: A Simulation-Based Analysis

The Beta distribution is a highly useful continuous probability distribution defined on the interval (0, 1) ‎and characterized by two shape parameters, α and β. Accurate estimation of these parameters is ‎crucial in various fields, ranging from reliability engineering to Bayesian A/B testing and click-through ‎rate smoothing in data science. This paper presents a comprehensive comparative analysis of three parameter estimation frameworks: the Method of Moments (MOM), Maximum Likelihood Estimation ‎‎(MLE), and Bayesian estimation via Maximum A posteriori (MAP) under a Gamma prior. Building on ‎previous investigations which demonstrated the superiority of MOM in small sample sizes (N<50), ‎this study conducts extensive Monte Carlo simulations across multiple scenarios (symmetric, skewed, ‎and U-shaped distributions) and sample sizes ranging from 10 to 500. Our results confirm that MOM remains computationally trivial and has lower mean squared error (MSE) than MLE in extremely small ‎samples (N=10), while MLE is asymptotically efficient and excels in large samples. The Bayesian ‎MAP estimator with a Gamma(2, 0.5) prior outperforms MLE in small-to-moderate samples in the ‎skewed and symmetric scenarios, with MSE reductions of roughly 90% in the most favourable case, and ‎converges to MLE as the sample size increases. A sensitivity analysis over six Gamma priors shows, ‎however, that this gain depends on how well the prior is located relative to the true parameters: poorly ‎located priors give little improvement or are worse than MLE, most clearly in the U-shaped scenario. Convergence diagnostics over 21,000 simulated datasets show no optimizer failures or boundary solutions for MLE or MAP and no variance violations for MOM. We discuss these trade-offs and provide ‎concrete recommendations for practitioners.