Scoring algorithm, also known as Fisher's scoring, is a form of Newton's method used in statistics to solve maximum likelihood equations numerically, named after Ronald Fisher.
Let Y 1 , … , Y n be random variables, independent and identically distributed with twice differentiable p.d.f. f ( y ; θ ) , and we wish to calculate the maximum likelihood estimator (M.L.E.) θ ∗ of θ . First, suppose we have a starting point for our algorithm θ 0 , and consider a Taylor expansion of the score function, V ( θ ) , about θ 0 :
V ( θ ) ≈ V ( θ 0 ) − J ( θ 0 ) ( θ − θ 0 ) , where
J ( θ 0 ) = − ∑ i = 1 n ∇ ∇ ⊤ | θ = θ 0 log f ( Y i ; θ ) is the observed information matrix at θ 0 . Now, setting θ = θ ∗ , using that V ( θ ∗ ) = 0 and rearranging gives us:
θ ∗ ≈ θ 0 + J − 1 ( θ 0 ) V ( θ 0 ) . We therefore use the algorithm
θ m + 1 = θ m + J − 1 ( θ m ) V ( θ m ) , and under certain regularity conditions, it can be shown that θ m → θ ∗ .
In practice, J ( θ ) is usually replaced by I ( θ ) = E [ J ( θ ) ] , the Fisher information, thus giving us the Fisher Scoring Algorithm:
θ m + 1 = θ m + I − 1 ( θ m ) V ( θ m ) .
