Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 29J / a / Solution

Letbe the log-likelihood. The score function and scalar Fisher information areUnder the usual regularity conditions the score has mean zero, so this is also its variance.

The information tensorizes when it is additive over observations. For identically distributed observations,For an independent sample the joint score is the sum of the individual scores, and the mean-zero score identity makes all cross covariances vanish. This is the Tensorization of Fisher information.