Formation matrix
In statistics and information theory, the expected formation matrix of a likelihood function formula_1 is the matrix inverse of the Fisher information matrix of formula_1, while the observed formation matrix of formula_1 is the inverse of the observed information matrix of formula_1. Currently, no notation for dealing with formation matrices is widely used, but in books and articles by Ole E. Barndorff-Nielsen and Peter McCullagh, the symbol formula_5 is used to denote the element of the i-th line and j-th column of the observed formation matrix. The geometric interpretation of the Fisher information matrix (metric ) leads to a notation of formula_6 following the notation of the (contravariant) metric tensor in differential geometry. The Fisher information metric is denoted by formula_7 so that using Einstein notation we have formula_8. These matrices appear naturally in the asymptotic expansion of the distribution of many statistics related to the likelihood ratio.