Scoring algorithm, also known as Fisher's scoring,[1] is a form of Newton's method used in statistics to solve maximum likelihood equations numerically, named after Ronald Fisher.

Sketch of derivation

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Let   be random variables, independent and identically distributed with twice differentiable p.d.f.  , and we wish to calculate the maximum likelihood estimator (M.L.E.)   of  . First, suppose we have a starting point for our algorithm  , and consider a Taylor expansion of the score function,  , about  :

 

where

 

is the observed information matrix at  . Now, setting  , using that   and rearranging gives us:

 

We therefore use the algorithm

 

and under certain regularity conditions, it can be shown that  .

Fisher scoring

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In practice,   is usually replaced by  , the Fisher information, thus giving us the Fisher Scoring Algorithm:

 ..

Under some regularity conditions, if   is a consistent estimator, then   (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.[2]

See also

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References

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  1. ^ Longford, Nicholas T. (1987). "A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects". Biometrika. 74 (4): 817–827. doi:10.1093/biomet/74.4.817.
  2. ^ Li, Bing; Babu, G. Jogesh (2019), "Bayesian Inference", Springer Texts in Statistics, New York, NY: Springer New York, Theorem 9.4, doi:10.1007/978-1-4939-9761-9_6, ISBN 978-1-4939-9759-6, S2CID 239322258, retrieved 2023-01-03

Further reading

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