# Is a comparison between Bayesian and frequentist prediction intervals sensible?

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.