I am fitting a linear regression model to 10 data points and then making posterior predictions on thousands of points. I am worried about understating the uncertainty of the predictions. One reason for this worry is the difference between the in-sample estimated residual standard deviation
sigma from the LOO residual standard deviation
sigma_loo with the latter being about 25% larger than the former and outside of its posterior 95% interval.
I could try to replace (or scale)
sigma by (or with)
sigma_loo but this seems crude and somewhat incoherent from a Bayesian perspective. Is there a standard way to achieve my goal of incorporating the LOO information about the posterior uncertainty into the posterior predictive? Or an argument for why the original
sigma is adequate?