I am estimating a Bayesian multiple regression using continuous data on both the dependent variable and the regressors. My goal is to iteratively estimate the coefficient distributions as more data becomes available, each time using the posterior distribution of the last estimation as the new prior.
I am using Normal priors for the regressor coefficients and the intercept and a Half Normal prior for sigma (the regression error). I realize that, assuming normal likelihood for the dependent variable, the posteriors for the regression coefficients are Normal as well.
I am using Gaussian kernel density estimation to approximate their posterior distributions and feed them as priors for the next iteration. My question is how I should do this for sigma, given that I don't know the form of its posterior distribution.
If anyone has a suggestion involving a different prior for sigma (e.g. Half Cauchy or Inverse Gamma), that would also be fine.
Thanks in advance!