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2022

von Mises-Fisher distributions and their statistical divergence

Kitagawa, Toru, Rowley, Jeff

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The von Mises-Fisher family is a parametric family of distributions on the surface of the unit ball, summarised by a concentration parameter and a mean direction.

As a quasi-Bayesian prior, the von Mises-Fisher distribution is a convenient and parsimonious choice when parameter spaces are isomorphic to the hypersphere (e.g., maximum score estimation in semi-parametric discrete choice, estimation of single-index treatment assignment rules via empirical welfare maximisation, under-identifying linear simultaneous equation models).
Despite a long history of application, measures of statistical divergence have not been analytically characterised for von Mises-Fisher distributions.
This paper provides analytical expressions for the $f$-divergence of a von Mises-Fisher distribution from another, distinct, von Mises-Fisher distribution in $\mathbb{R}^p$ and the uniform distribution over the hypersphere.

von Mises-Fisher distributions and their statistical divergence · Around

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von Mises, R
1918
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Fisher, R. A
1953
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Abramowitz, M. and I. A. Stegun
1964
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Pinsker, M. S
1964
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Amos, D. E
1974
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Bingham, C
1974
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Manski, C
1975
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Mardia, K. V
1975
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Mardia, K. V. and S. El-Atoum
1976
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Bretagnolle, J. and C. Huber
1978
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Best, D. and N. I. Fisher
1979
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Kent, J. T
1982
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Watson, G. S
1984
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