Bayesian approach to cubic natural exponential families

TL;DR

This paper extends Bayesian conjugate prior theory from quadratic to cubic natural exponential families, introducing new classes of priors and characterizations for these more complex distributions.

Contribution

It introduces a new class of NEFs and associated conjugate priors, extending Bayesian properties from quadratic to cubic NEFs.

Findings

Extended Bayesian properties to cubic NEFs

Defined new conjugate prior families for these NEFs

Characterized cubic NEFs using Bayesian methods

Abstract

For a natural exponential family (NEF), one can associate in a natural way two standard families of conjugate priors, one on the natural parameter and the other on the mean parameter. These families of conjugate priors have been used to establish some remarkable properties and characterization results of the quadratic NEF's. In the present paper, we show that for a NEF, we can associate a class of NEF's, and for each one of these NEF's, we define a family of conjugate priors on the natural parameter and a family of conjugate priors on the mean parameter which are different of the standard ones. These families are then used to extend to the Letac-Mora class of real cubic natural exponential families the properties and characterization results related to the Bayesian theory established for the quadratic natural exponential families.