Bayesian Hierarchical Modeling for Parameter Estimation in Normal Distribution with Aggregated Data

This study addresses the challenge of estimating statistical parameters from individual observations based on aggregated data, a task of paramount importance in scenarios where direct access to raw data is restricted by privacy concerns, logistical constraints, or storage limitations. Drawing inspiration from recent research that assumes a normal distribution for individual observations to derive maximum likelihood estimates for mean and standard deviation, we confront the limitations inherent in classical approaches, especially under questionable normality assumptions or when dealing with intractable likelihood functions. To circumvent these issues, we turn to Bayesian methods, which offer enhanced flexibility in model specification and allow for the incorporation of prior knowledge and uncertainty into the analysis. Within the Bayesian framework, we employ Markov Chain Monte Carlo (MCMC) techniques to effectively sample from the posterior distribution when direct calculation is infeasible. Additionally, for situations where the likelihood function is difficult to specify or evaluate, we explore the use of Approximate Bayesian Computation (ABC) as a pragmatic alternative. Our study conducts a comparative analysis of these methodologies, emphasizing their unique strengths and their suitability for different analytical scenarios.