{
  "id": "enwiki-001360734777",
  "slug": "scoring-algorithm",
  "title": "Scoring algorithm",
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  "lead": "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.",
  "body": "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. ==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{J}(\\theta_0)(\\theta - \\theta_0), \\, where \\mathcal{J}(\\theta_0) = - \\sum_{i=1}^n \\left. \\nabla \\nabla^{\\top} \\right|_{\\theta=\\theta_0} \\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_{0} + \\mathcal{J}^{-1}(\\theta_{0})V(\\theta_{0}). \\, We therefore use the algorithm \\theta_{m+1} = \\theta_{m} + \\mathcal{J}^{-1}(\\theta_{m})V(\\theta_{m}), \\, and under certain regularity conditions, it can be shown that \\theta_m \\rightarrow \\theta^* . ==Fisher scoring== In practice, \\mathcal{J}(\\theta) is usually replaced by \\mathcal{I}(\\theta)= \\mathrm{E}[\\mathcal{J}(\\theta)] , the Fisher information, thus giving us the Fisher Scoring Algorithm: \\theta_{m+1} = \\theta_{m} + \\mathcal{I}^{-1}(\\theta_{m})V(\\theta_{m}) .. Under some regularity conditions, if \\theta_m is a consistent estimator, then \\theta_{m+1} (the correction after a single step) is 'optimal' in the sense that its error distribution is asymptotically identical to that of the true max-likelihood estimate. ==See also== Score (statistics) Score test Fisher information ==References== ==Further reading==",
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    "title": "Scoring algorithm",
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        "family": "Anticloud FZ LLE",
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    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/scoring-algorithm",
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      "date-parts": [
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          2026,
          11,
          12
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