# Formation matrix

In statistics and information theory, the expected formation matrix of a likelihood function $L(\theta)$ is the matrix inverse of the Fisher information matrix of $L(\theta)$, while the observed formation matrix of $L(\theta)$ is the inverse of the observed information matrix of $L(\theta)$.[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 $j^{ij}$ 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 $g^{ij}$ following the notation of the (contravariant) metric tensor in differential geometry. The Fisher information metric is denoted by $g_{ij}$ so that using Einstein notation we have $g_{ik}g^{kj} = \delta_i^j$.

These matrices appear naturally in the asymptotic expansion of the distribution of many statistics related to the likelihood ratio.