I am aware that frequentist confidence intervals and Bayesian credible intervals have quite different interpretations, and are not comparable.

I'm wondering if the same is true for prediction intervals. A frequentist prediction interval is an interval $(l,u)$ such that $Pr(l < y^{new} < u) = 1 - \alpha$ for a predetermined $\alpha$. The definition of a Bayesian prediction interval is ... the same? I believe in both methodologies, one is permitted to say "there is a $100\times(1-\alpha)\%$ chance the next observation $y^{new}$ lies in $(l, u)$" because $y^{new}$ is random.

Given a model, say simple linear regression, $y_i = a + bx_i$, I would like to compare the empirical coverage probability (from simulation) of a frequentist prediction interval for $y^{new}|x$ with the coverage probability of the highest density interval calculated on the posterior predictive distribution of $y^{new}|x$. I think this is permissible because the intervals have the same interpretation, but I could be way off.


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