Marginally Useful: An Information-Gap Identity in Conformal Prediction

Conformal prediction has been touted as a more formal, rigorous approach to adding uncertainty to a forecast. The sole objective of this note is to point out that rigor cuts both ways in the case of residual pooling, the technique used in the vast majority of conformal prediction applications. The fact that unconditional guarantee of coverage is provided is not in question, but we make clear, we believe for the first time, that there is an opposing guarantee too: a permanent gambit of logarithmic-score regret which no amount of data or tuning can subsequently reduce. We give the exact size of the sacrifice, and a financial reading of it as the growth rate of an oracle adversary betting against odds set by someone using residual pooling.