Scoring algorithm

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.

References

References

1. Longford, Nicholas T.. (1987). "A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects". Biometrika.
2. (2019). "Bayesian Inference". Springer New York.