@jonah and I talk about this from time to time.

Itâ€™s kind of a funny thing because in frequentist statistics more often than not you see confidence intervals reported. Prediction intervals seem much less common so itâ€™s surprising that Bayesian inference is (at least as far as I know) entirely prediction intervals.

Rob Hyndman has a nice blog post elaborating why that is (https://robjhyndman.com/hyndsight/intervals/), but the cliffâ€™s notes are that once the parameter is modeled as a random variable, then inherent meaning of a confidence interval (i.e., 95% of all 95% confidence intervals contain the true parameter value) doesnâ€™t make sense.

Some people (including me!) do sometimes look a frequentist properties of Bayesian intervals. That can be contentious but I view it as calibration of methods in a very practical sense.