NORMAL-NORMAL POSTERIOR ESTIMATION OF LINEAR REGRESSION PARAMETERS USING QUANTILES: A MONTE CARLO APPROACH

Abstract

This research work focused on Bayesian estimation of linear regression to determine the quantile range at which optimal hyperparameters of normally distributed data with vague information could be obtained. This is a buildup on the study by Fasoranbaku et al (2015) which focused on obtaining Bayesian posterior estimates using quantiles. In this study, normally distributed data sets of sample size 100 were generated through Monte Carlo simulation approach. Ordinary Least Squares analysis was run on each data set to obtain the Confidence Intervals (CI) for the regression parameters and their variances which were then divided into 10 and 100 equal parts to obtain the hyperparameters of the prior distribution. Observation precisions, Posterior precisions were also estimated from the regression output to determine the Posterior means (Bayesian estimates)  0 -  2 for each of the 9/99 models to derived the new dependent variables (Y). Mean Absolute Deviation (MAD) was used to determine the adequacy of each model to choose the best model from the 9/99 models. This process was repeated 1000 times to determine the quantile range for the optimal hyperparameters. The study revealed that the optimal hyperparameters are located mostly between the 4th and 6th deciles when the confidence intervals were divided into 9 grid points and between 34th and 66th percentiles when they were divided into 99 grid points. The study, however, explored the possibility of obtaining prior distribution from the data with vague information, revealed the optimal quantile range where optimal hyperparameters that produce better model in regression analysis could be found and the superiority of Bayesian linear regression to OLS when Mean Absolute Deviation (MAD) is employed as the model adequacy criterion.