Polynomial Exponential Families (PEFs) are exponential families with sufficient statistics a set of monomials. When there is a single monomial sufficient statistics, PEFs are called Monomial Exponential Families (MEFs).
Work:

• Fast approximations of the Jeffreys divergence between univariate Gaussian mixtures via mixture conversions to exponential-polynomial distributions, Entropy 23.11 (2021): 1417. (technical report 2107.05901)
• Patch matching with polynomial exponential families and projective divergences, International Conference on Similarity Search and Applications. Cham: Springer International Publishing, 2016.
• MaxEnt upper bounds for the differential entropy of univariate continuous distributions, IEEE Signal Processing Letters 24.4 (2017): 402-406. (technical report 1612.02954)
• Statistical exponential families: A digest with flash cards, arXiv:0911.4863

References:

• Cobb, Loren, Peter Koppstein, and Neng Hsin Chen. "Estimation and moment recursion relations for multimodal distributions of the exponential family." Journal of the American Statistical Association 78.381 (1983): 124-130.
• Mead and Papanicolaou, "Maximum entropy in the problem of moments." Journal of Mathematical Physics 25.8 (1984): 2404-2417.

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July 2023