Conjugate Priors for Exponential Families Having Cubic Variance Functions
نویسندگان
چکیده
In this paper, we give three equivalent properties of the class of multivariate simple cubic natural exponential families (NEF’s). The first property says that the cumulant function of any basis of the family is a solution of some Monge-Ampère equation, the second is that the variance function satisfies a differential equation, and the third is characterized by the equality between two families of prior distributions related to the NEF. These properties represent the extensions to this class of the properties stated in [1] and satisfied by the Wishart and the simple quadratic NEF’s. We also show that in the real case, each of these properties provides a new characterization of the Letac-Mora class of real cubic NEF’s.
منابع مشابه
Bayesian approach to cubic natural exponential families
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...
متن کاملTransorthogonal polynomials and simple cubic multivariate distributions
Transorthogonality for a sequence of polynomials on Rd has been recently introduced in order to characterize the reference probability measures, which are multivariate distributions of the natural exponential families (NEFs) having a simple cubic variance function. The present paper pursues this characterization of three various manners through exponential generating functions, transdiagonality...
متن کاملGibbs Sampling, Conjugate Priors and Coupling
We give a large family of simple examples where a sharp analysis of the Gibbs sampler can be proved by coupling. These examples involve standard statistical models – exponential families with conjugate priors or location families with natural priors. Many of them seem difficult to succesfully analyze using spectral or Harris recurrence techniques.
متن کاملInvariant conjugate analysis for exponential families
There are several ways to parameterize a distribution belonging to an exponential family, each one leading to a different Bayesian analysis of the data under standard conjugate priors. To overcome this problem, we propose a new class of conjugate priors which is invariant with respect to smooth reparameterization. This class of priors contains the Jeffreys prior as a special case, according to ...
متن کاملGibbs Sampling, Exponential Families and Coupling
We give examples of a quantitative analysis of the bivariate Gibbs sampler using coupling arguments. The examples involve standard statistical models – exponential families with conjugate priors or location families with natural priors. Our main approach uses a single eigenfunction (always explicitly available in the examples in question) and stochastic monotonicity.
متن کامل