Closed form posteriors for a simple bivariate Bayesian regression

Question

I'm analyzing a simple linear regression $Y_{i}$~$a+b*X_{i}+e_{i}$, with $e$ being normally distributed with known variance and where I have normal priors on $a$ and $b$. I'm trying to piece together an approximate closed form solution for the variance of my posterior of $a$ as a function of sample size for the purposes of an optimization I'm trying to do.

I know that in general, there's a closed form conjugate prior for Bayesian regression in the multivariate case in matrix form. So I can do some sort of moment matching and then calculate out the matrix algebra for the bivariate case and probably can end up with a nice looking closed form. I've tried that, but I keep getting bogged in algebra, and I feel like this has to be a well-known problem that's been fleshed out already. Does anybody have a citation or link to point me to?

Answer 1

Please, take a look at Section 3.2.6 of the following document:

http://www.biostat.uzh.ch/teaching/master/previous/seminarbayes/SimonKunz_article.pdf

There, you will find the posterior of $(a,b)$ (which is a bivariate normal distribution in your case with a certain known variance structure) from where you can extract the marginal distributions, which are normal, and derive the corresponding variances.

Also, page 19 of the following document:

http://fisher.osu.edu/~schroeder.9/AMIS900/ech6.pdf

References

Bürgin, Reto: Bayesian linear model - Basics, Part 1. 2009

Gelman et al.: Bayesian data analysis, third edition.