Prediction interval vs Confidence interval

Rob Hyndman has a blog post where he details the difference between confidence interval, prediction interval, and credible interval.

I think he makes an good point that confidence interval and prediction interval are often used inter-changeably, even they are quite different. Confidence interval comes from the realm of frequentist inference, and applies to a parameter that has been evaluated using a statistical method. A 95% confidence interval is an interval that will contain the true value of that parameter 95% of the time, if we could repeat the experiment indefinitly.

A prediction interval is an interval for a predicted value. Not for a model parameter. That predicted value does not exist yet (that’s why we need to predict it), and its uncertainty is represented by a random variable. A 70% prediction interval will contain 70% of the mass of that random variable; or to be more general, this would be the 70% HDI.

A credibility interval is kind of a mix between the two. It’s the Bayesian equivalent of a confidence interval, and therefore applies to the posterior distribution of a model parameter. A 75% credibility interval is the 75% HDI of the posterior distribution.