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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 distance function should be independent of the choice of parameterization of the family of conditional pdfs and...
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References
Amari S-I (1985) Differential-geometric methods in statistics. Lecture notes in statistics, vol. 28. Springer, New York
Fisher RA (1922) On the mathematical foundations of theoretical statistics. Phil Trans R Soc London Ser A 222:309–368
Maybank SJ (2004) Detection of image structures using Fisher Information and the Rao metric. IEEE Trans Pattern Anal Mach Intell 26(12):49–62
Mio W, Liu X (2006) Landmark representations of shapes and Fisher-Rao geometry. In: Proceedings of the IEEE international conference on image processing, Atlanta. IEEE, pp 2113–2116
Peter A, Rangarajan A (2009) Information geometry for landmark shape analysis: unifying shape representation and deformation. IEEE Trans Pattern Anal Mach Intell 31(2):337–350
Rao C (1945) Information and accuracy attainable in the estimation of statistical parameters. Bull Calcutta Math Soc 37:81–91
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Maybank, S.J. (2014). Fisher-Rao Metric. In: Ikeuchi, K. (eds) Computer Vision. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-31439-6_657
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DOI: https://doi.org/10.1007/978-0-387-31439-6_657
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Publisher Name: Springer, Boston, MA
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