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Conditionally reducible natural exponential families and enriched conjugate priors. (English) Zbl 0973.62011
The authors consider multidimensional natural exponential families (NEF) of distributions and introduce the notion of conditionally reducible NEF (cr-NEF), i.e., a NEF whose density can be represented as a product of NEFs’ densities at lower dimensions (components). (Note that these densities are conditional for fixed “previous” observations, so they can be functions of these observations).
A structure of cr-NEF is described and conditions of conditional reducibility are given. For cr-NEFs the notion of enriched standard conjugate (ESCF) family is introduced. ESCF is not a standard conjugate prior for the NEF, but rather a product of independent standard conjugate priors of its components. So such families are richer than the standard conjugate priors. As an application, NEFs with simple quadratic variance functions are considered.

##### MSC:
 62E10 Characterization and structure theory of statistical distributions 62F15 Bayesian inference
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