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.
