Fisher information: one construct, many systems

In mathematical statistics, Fisher information quantifies how much an observable random variable carries about an unknown parameter: it is the expected curvature (the variance of the score) of the log-likelihood, and larger information means the parameter can be estimated more precisely, with the Cramér–Rao bound setting the lowest variance any unbiased estimator can reach. Fisher's On the mathematical foundations of theoretical statistics introduced this measure of the information in a sample. The referent is the information content of data about a parameter, measured in the reciprocal-variance units that reappear, unchanged, in test theory and in experimental design.[1]

Unit

Expected curvature of the log-likelihood, bounding estimator variance

Recovery means

Not applicable

Measured by

Computed as the expected squared score (negative expected second derivative of the log-likelihood)

In item response theory, the information function is exactly Fisher information applied to the latent trait: each item contributes information about ability that depends on the trait level, and summing item information gives the test information function, whose reciprocal is the squared standard error of the estimated trait. Lord's Applications of item response theory to practical testing problems uses this to show where on the scale a test measures precisely and to build tests that target a chosen range. It is the same construct as statistical Fisher information, realised in the measurement system and read as measurement precision across the trait continuum.[2]

Unit

Information about the latent trait at each trait level; its reciprocal is the squared standard error

Recovery means

Not applicable

Measured by

Summed across items as the test information function, inverted to give the conditional standard error of measurement

In optimal experimental design, the same construct becomes the Fisher information matrix of a proposed design, and choosing where and how to collect data amounts to making that matrix as large as possible so parameter estimates come out as precise as possible, with criteria such as D- and A-optimality defined on it. Atkinson and Donev's Optimum experimental designs develops how designs are compared and built by maximising functions of this information matrix. It is the identical information construct as in general statistics and in test theory, here guiding the collection of data rather than describing data already in hand.[3]

Unit

The Fisher information matrix of a design, summarised by optimality criteria

Recovery means

Not applicable

Measured by

Optimised through criteria (D-, A-, E-optimality) defined on the design's information matrix