Conditionally reducible natural exponential families and enriched conjugate priors (Q2739870)

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).NEWLINENEWLINENEWLINEA 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.