Preamble Attenion Conservation Notice: I implement the now-standard Bayesian procedure for estimating a Gaussian mean and variance with weakly informative priors using Stan and make some connections to confidence distributions and fiducial inference. But without any of the details for this to make sense for a newcomer. For the former material, you are better served by page 73 of A First Course in Bayesian Statistical Methods by Peter Hoff or page 67 of Bayesian Data Analysis.