Bayesian Point and Lnterval Estimation of σ²

This research focuses on Bayesian estimation of variance and its application in statistical modeling. The study highlights the theoretical foundations of Bayesian inference, emphasizing the use of conjugate priors, particularly the Inverse-Gamma distribution, to estimate variance parameters in normally distributed data. The Bayesian framework combines prior information with observed data to generate posterior distributions, which provide both point estimates and probabilistic measures of uncertainty. A practical simulation was conducted using samples of varying sizes, demonstrating how posterior means and posterior modes (MAP) effectively estimate the true variance. The study also illustrates the computation and interpretation of Bayesian credible intervals, which offer a direct probability-based measure of uncertainty, in contrast to traditional confidence intervals. Results show that Bayesian estimates are accurate, stable, and converge toward classical estimates as the sample size increases, while retaining robustness for small or dispersed datasets. Overall, the research confirms that Bayesian variance estimation is a powerful and flexible tool for statistical analysis. It enhances the precision of parameter estimation, provides meaningful probabilistic interpretation of uncertainty, and can be readily applied to a wide range of applied and scientific problems where reliable variance estimation is critical.