In statistics and information theory, the expected formation matrix of a likelihood function
L
(
θ
)
is the matrix inverse of the Fisher information matrix of
L
(
θ
)
, while the observed formation matrix of
L
(
θ
)
is the inverse of the observed information matrix of
L
(
θ
)
.
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
i
j
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
i
j
following the notation of the (contravariant) metric tensor in differential geometry. The Fisher information metric is denoted by
g
i
j
so that using Einstein notation we have
g
i
k
g
k
j
=
δ
i
j
.
These matrices appear naturally in the asymptotic expansion of the distribution of many statistics related to the likelihood ratio.
