Autore
Pezzulli, Sergio

Titolo
Fitting hierarchical normal models
Periodico
Università degli Studi di Roma "La Sapienza" - Dipartimento di Studi Geoeconomici, Linguistici, Statistici e Storici per l'Analisi regionale. Working papers
Anno: 1999 - Fascicolo: 14 - Pagina iniziale: 1 - Pagina finale: 25

Multivariate craniometric data of the Etruscan collection constitute the motivating study. It exemplifies independent sampling from g Gaussian sub-populations with unknown mean vectors and precision matices P (reciprocal variances), j=1,2..,g. Contaminations are admitted by a simple mixture model. Also, the missing-at-random (MAR) hypothesis simplify the missing data case. A hierarchy is defined by a Bayesian model with a peculiar three-stages prior on the joint list of the parameters j=1,2..,g) and some additional unknowns (hyper-parameters). All the stages are given in the simplest form: the conjugate. At the first stages, the parameters are assumed to be an iid sample from a common prior. It is a Wishart-Normal distribution (WN) on dependent components P, with fixed hyperparameters. In two higher stages, these further unknowns are again subjected to randomization. The overall distributive (scale and center) tendencies of the sub-population parameters are defined to be a WN-variate in the Jeffreys uninformative position. The third stage prior, finally, regards the dispersion witheen the parameters. More precisely, the attraction witheen scales is fixed with probability one: a cautions (small) value can be suggested by maximum likelihood (ML) estimation. On the other hand certain vriance-inflating factors, regarding the mean parameters, are assumed instead to be a random sample from a given Gamma distribution. By the latter prior assumption, small-shrinking degrees are highly probable, but data can change this expectation. In particular, the overall unknowns of the hierarchy can be jointly estimated by a posterior mode. To this aim a conditional maximization algotithm is derived. Among the results, the fitted parameters can be seen like posterior corrections to the g separate ML solutions. The algorithm is formed of known and new steps, easily offered by conjugacy and Bayes theorem




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