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Scoring algorithm : ウィキペディア英語版
Scoring algorithm

Scoring algorithm, also known as Fisher's scoring,〔(A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects )〕 is a form of Newton's method used in statistics to solve maximum likelihood equations numerically, named after Ronald Fisher.
==Sketch of Derivation==
Let Y_1,\ldots,Y_n be random variables, independent and identically distributed with twice differentiable p.d.f. f(y; \theta), and we wish to calculate the maximum likelihood estimator (M.L.E.) \theta^
* of \theta. First, suppose we have a starting point for our algorithm \theta_0, and consider a Taylor expansion of the score function, V(\theta), about \theta_0:
: V(\theta) \approx V(\theta_0) - \mathcal(\theta_0)(\theta - \theta_0), \,
where
: \mathcal(\theta_0) = - \sum_^n \left. \nabla \nabla^ \right|_ \log f(Y_i ; \theta)
is the observed information matrix at \theta_0. Now, setting \theta = \theta^
*, using that V(\theta^
*) = 0 and rearranging gives us:
: \theta^
* \approx \theta_ + \mathcal^(\theta_)V(\theta_). \,
We therefore use the algorithm
: \theta_ = \theta_ + \mathcal^(\theta_)V(\theta_), \,
and under certain regularity conditions, it can be shown that \theta_m \rightarrow \theta^
*.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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