Applications of Bayesian statistics
Facing challenges like designing complex models, summarizing large data sets and fitting probability models, applied statisticians frequently appeal to using commercial software packages. A majority of the packages use a classical or a frequentist statistical process. Bayesian Analysis, an alternative to the classical one, comes as a rather effective way to deal with statistical problems and it has many applications. Fitting a simple linear regression model is commonly solved by estimating the parameters by the Least Square Estimation method. One limitation of the Least Square Estimation method comes when looking at how to assess the effect of covariates on a specific quantile of the dependent variables. Quantile Regression is used in that respect and it is a convenient way to perform linear regression by estimating the conditional quantiles. When we think about the precision of our estimates, the Bayesian approach of Quantile Regression has one advantage in that it is possible to have the exact posterior distribution of the conditional coefficients of the regression model. E.Tsionas  gives an effective approach to obtain the posterior probabilities from the assumption of a scale mixture of the normal distribution on the error term of the linear model. The posterior distribution of the parameter of interest can be obtained using data augmentation around Gibbs Sampling with WinBUGS. In that case, Bayesian Median regression seems to be easily tractable - it remains a challenge to obtain the distribution of other specific quantiles. In this thesis, we will first start with Bayesian Median regression, and continue by segmenting the conditional response variable which would have in this case a conditional distribution. Using R which is a convenient environment to implement with the package BRugs and after a limited number of estimations, we will derive a regression model on those parameters estimates in order to infer about the estimated ones.