Score Function

The score function is the partial derivative of the log-likelihood function , where is the standard likelihood function.

For independent observations with densities , the likelihood function

(1)

gives

(2)

and thus that

			(3)
			(4)
			(5)

Using the above formulation of , one can easily compute various statistical measurements associated with . For example, the mean of is denoted , where denotes the expectation value of . Under regularity assumptions, including parameter-independent support and justification of differentiation under the integral, this expectation value is zero and the variance equals the Fisher information matrix (a scalar for a scalar parameter). Without these assumptions the identity can fail. For a uniform observation on , the score is almost everywhere, so its expectation value is and its variance is zero. The score function has extensive uses in many areas of mathematics, both pure and applied, and is a key component of the field of likelihood theory.