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Bregman divergences based on optimal design criteria and simplicial measures of dispersion

Author

Listed:
  • Luc Pronzato

    (CNRS, UCA, Laboratoire I3S, UMR 7172; 2000, route des Lucioles, Les Algorithmes)

  • Henry P. Wynn

    (London School of Economics)

  • Anatoly Zhigljavsky

    (Cardiff University)

Abstract

In previous work the authors defined the k-th order simplicial distance between probability distributions which arises naturally from a measure of dispersion based on the squared volume of random simplices of dimension k. This theory is embedded in the wider theory of divergences and distances between distributions which includes Kullback–Leibler, Jensen–Shannon, Jeffreys–Bregman divergence and Bhattacharyya distance. A general construction is given based on defining a directional derivative of a function $$\phi $$ ϕ from one distribution to the other whose concavity or strict concavity influences the properties of the resulting divergence. For the normal distribution these divergences can be expressed as matrix formula for the (multivariate) means and covariances. Optimal experimental design criteria contribute a range of functionals applied to non-negative, or positive definite, information matrices. Not all can distinguish normal distributions but sufficient conditions are given. The k-th order simplicial distance is revisited from this aspect and the results are used to test empirically the identity of means and covariances.

Suggested Citation

  • Luc Pronzato & Henry P. Wynn & Anatoly Zhigljavsky, 2019. "Bregman divergences based on optimal design criteria and simplicial measures of dispersion," Statistical Papers, Springer, vol. 60(2), pages 545-564, April.
    • Handle: RePEc:spr:stpapr:v:60:y:2019:i:2:d:10.1007_s00362-018-01082-8
    DOI: 10.1007/s00362-018-01082-8
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    References listed on IDEAS

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    1. Luc Pronzato & Henry P. Wynn & Anatoly Zhigljavsky, 2016. "Extremal measures maximizing functionals based on simplicial volumes," Statistical Papers, Springer, vol. 57(4), pages 1059-1075, December.
    2. Pronzato, Luc & Wynn, Henry P. & Zhigljavsky, Anatoly A., 2018. "Simplicial variances, potentials and Mahalanobis distances," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 276-289.
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    Cited by:

    1. Elham Yousefi & Luc Pronzato & Markus Hainy & Werner G. Müller & Henry P. Wynn, 2023. "Discrimination between Gaussian process models: active learning and static constructions," Statistical Papers, Springer, vol. 64(4), pages 1275-1304, August.

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