I am a graduate student in math working in number theory. However, I am taking a course on Stochastic optimization this semester. I have been assigned a project on the course which I describe below along with the ideas that I have in mind about it. I would appreciate if I can get any expert advice on how to actually go about the problem since it's specifically related to probability and statistics and I don't much work on these areas.
Here is my problem. I have been given a set of data in the form (x-values, y-values)=$\{(x_i,y_i): 1\leq i\leq \mbox{(some known positive integer, say 40)}\},$ where $x$ and $y$ are independent and dependent variables respectively. I want to model the probability distribution of $y$ using Bayesian inference.
I have a basic idea of bayesian statistics, however I was wondering how I would use the given data to get any (if at all) probabilistic information out of it. A way to use it is by considering the logarithm of the joint density of $x$ and $y$ of the data as described in this article http://www.ism.ac.jp/editsec/aism/pdf/055_1_0001.pdf. It explains the problem that I am dealing with, however I was wondering if there is a simpler idea to do the same.
I know an alternative approach to do it is by using linear regression using a normal distribution for the error. Another way of doing it is by quantile regression. However, I am looking for a simpler bayesian approach using the given data set.
I would appreciate any suggestions/references that you would like to provide in this regard.