Prediction interval = credible interval?

Question

I am wondering if prediction interval and credible interval evaluate the same thing.

For instance with a linear regression, when you estimate the prediction interval of a fitted values, you estimate the $(1-\alpha)\%$ limits of the interval in which you expect your value to fall. Conversely to a confidence interval, you do not focus on a distribution parameter such as the mean value, but on the value that your explained variable could take for a given X value (supposing that $\ Y = a + b.X$).

When you estimate the fitted value for a given $X$ value within a Bayesian framework, from the posterior probability distribution, you can estimate a credible interval. Does this interval give you the same information on the fitted value or not?

Answer 1

They live in different spaces and mean different things.

A credible interval $[a,b]$ is a subset of the parameter space such that $$ P(a\leq\Theta\leq b\mid X_1=x_1,\dots,X_n=x_n) = \alpha \, , $$ and it means that, after seeing the data, you believe that with probability $\alpha$ the parameter value is inside this interval.

A prediction interval $[u,v]$ is a subset of the sampling space such that $$ P(u\leq X_{n+1}\leq v\mid X_1=x_1,\dots,X_n=x_n) = \gamma \, , $$ and it means that, after seeing the data, you believe that with probability $\gamma$ the value of a future observation $X_{n+1}$ will be inside this interval.