Remark. The covariance is predictive only for a declared reward [ftip-008R]
Remark. The covariance is predictive only for a declared reward [ftip-008R]
The identity Theorem [ftip-008Q] expresses exact tilted-law gain using expectations under \(p\). This makes the population quantity accessible from the reference law in principle. A finite-sample estimator still needs its own sampling law, moment assumptions, and error analysis.
The formula also remains indexed by both \(r\) and \(s\). It cannot turn a proxy reward into an independent utility. If \(s=r\), it measures improvement in the same quantity that defined the exact optimizer; if \(s\ne r\), its sign is determined by the displayed covariance.