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# Fisher-Rao Metric

## Synonyms

Rao Metric

## Related Concepts

Cramér-Rao Lower Bound; Fisher Information; Hough Transform; Kullback-Leibler Divergence; Maximum Likelihood Estimation; Riemannian Metric

## Definition

The Fisher-Rao metric is a particular Riemannian metric defined on a parameterized family of conditional probability density functions (pdfs). If two conditional pdfs are near to each other under the Fisher-Rao metric, then the square of the distance between them is approximated by twice the average value of the log likelihood ratio of the conditional pdfs.

## Background

Suppose that a parameterized family of conditional pdfs is given and it is required to find the parameter value corresponding to the conditional pdf that best fits a given set of data. It is useful to have a distance function defined on pairs of conditional pdfs, such that if a given conditional pdf is a close fit to the data, then all the conditional pdfs near to it are also close fits to the data. Any such