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Conditionally Reducible Natural Exponential Families and Enriched Conjugate Priors

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  • Guido Consonni
  • Piero Veronese

Abstract

Consider a standard conjugate family of prior distributions for a vector‐parameter indexing an exponential family. Two distinct model parameterizations may well lead to standard conjugate families which are not consistent, i.e. one family cannot be derived from the other by the usual change‐of‐variable technique. This raises the problem of finding suitable parameterizations that may lead to enriched conjugate families which are more flexible than the traditional ones. The previous remark motivates the definition of a new property for an exponential family, named conditional reducibility. Features of conditionally‐reducible natural exponential families are investigated thoroughly. In particular, we relate this new property to the notion of cut, and show that conditionally‐reducible families admit a reparameterization in terms of a vector having likelihood‐independent components. A general methodology to obtain enriched conjugate distributions for conditionally‐reducible families is described in detail, generalizing previous works and more recent contributions in the area. The theory is illustrated with reference to natural exponential families having simple quadratic variance function.

Suggested Citation

  • Guido Consonni & Piero Veronese, 2001. "Conditionally Reducible Natural Exponential Families and Enriched Conjugate Priors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(2), pages 377-406, June.
  • Handle: RePEc:bla:scjsta:v:28:y:2001:i:2:p:377-406
    DOI: 10.1111/1467-9469.00243
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    Cited by:

    1. Oda, Hidemasa & Komaki, Fumiyasu, 2023. "Enriched standard conjugate priors and the right invariant prior for Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    2. Yanagimoto, Takemi & Ohnishi, Toshio, 2005. "Extensions of the conjugate prior through the Kullback-Leibler separators," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 116-133, January.
    3. Piero Veronese & Eugenio Melilli, 2015. "Fiducial and Confidence Distributions for Real Exponential Families," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(2), pages 471-484, June.
    4. Consonni, Guido & Massam, Hélène, 2012. "Parametrizations and reference priors for multinomial decomposable graphical models," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 380-396.
    5. Ip, Edward H. & Wang, Yuchung J., 2008. "A note on cuts for contingency tables," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2356-2363, November.
    6. Consonni, Guido & Veronese, Piero & Gutiérrez-Peña, Eduardo, 2004. "Reference priors for exponential families with simple quadratic variance function," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 335-364, February.

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