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?