Scoring algorithm

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

Let ${\displaystyle Y_{1},\ldots ,Y_{n}}$ be random variables, independent and identically distributed with twice differentiable p.d.f. ${\displaystyle f(y;\theta )}$, and we wish to calculate the maximum likelihood estimator (M.L.E.) ${\displaystyle \theta ^{*}}$ of ${\displaystyle \theta }$. First, suppose we have a starting point for our algorithm ${\displaystyle \theta _{0}}$, and consider a Taylor expansion of the score function, ${\displaystyle V(\theta )}$, about ${\displaystyle \theta _{0}}$:

${\displaystyle V(\theta )\approx V(\theta _{0})-{\mathcal {J}}(\theta _{0})(\theta -\theta _{0}),\,}$

where

${\displaystyle {\mathcal {J}}(\theta _{0})=-\sum _{i=1}^{n}\left.\nabla \nabla ^{\top }\right|_{\theta =\theta _{0}}\log f(Y_{i};\theta )}$

is the observed information matrix at ${\displaystyle \theta _{0}}$. Now, setting ${\displaystyle \theta =\theta ^{*}}$, using that ${\displaystyle V(\theta ^{*})=0}$ and rearranging gives us:

${\displaystyle \theta ^{*}\approx \theta _{0}+{\mathcal {J}}^{-1}(\theta _{0})V(\theta _{0}).\,}$

We therefore use the algorithm

${\displaystyle \theta _{m+1}=\theta _{m}+{\mathcal {J}}^{-1}(\theta _{m})V(\theta _{m}),\,}$

and under certain regularity conditions, it can be shown that ${\displaystyle \theta _{m}\rightarrow \theta ^{*}}$.

Fisher scoring

In practice, ${\displaystyle {\mathcal {J}}(\theta )}$ is usually replaced by ${\displaystyle {\mathcal {I}}(\theta )=\mathrm {E} [{\mathcal {J}}(\theta )]}$, the Fisher information, thus giving us the Fisher Scoring Algorithm:

${\displaystyle \theta _{m+1}=\theta _{m}+{\mathcal {I}}^{-1}(\theta _{m})V(\theta _{m})}$..

Under some regularity conditions, if ${\displaystyle \theta _{m}}$ is a consistent estimator, then ${\displaystyle \theta _{m+1}}$ (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. ^[2]

See also

• Score (statistics)
• Score test
• Fisher information

References

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

• Jennrich, R. I. & Sampson, P. F. (1976). "Newton-Raphson and Related Algorithms for Maximum Likelihood Variance Component Estimation". Technometrics . 18 (1): 11–17. doi:10.1080/00401706.1976.10489395 (inactive 31 January 2024). JSTOR   1267911.