Bayesian Analysis of the Independent Multinormal Process. Neither Mean Nor Precision Known

Abstract

SUMMARYUnder the assumption that neither the mean vector nor the variancecovariance matrix are known with certainty, the natural conjugate family of prior densities for the multivariate Normal process is identified.Prior-posterior and preposterior analysis is done assuming that the prior is in the natural conjugate family. A procedure is presented for obtaining non-degenerate joint posterior and preposterior distributions of all parameters even when the number of objective sample observations is less than the number of parameters of the process.

Mathematical Properties of the Multivariate t Distribution

“…In Bayesian analyses, (1) arises as: (a) the posterior distribution of the mean of a multivariate normal distribution [22,51]; (b) the marginal posterior distribution of the regression coefficient vector of the traditional multivariate regression model [54]; (c) the marginal prior distribution of the mean of a multinormal process [4]; (d) the marginal posterior distribution of the mean and the predictive distribution of a future observation of the multivariate normal structural model [20]; (e) an approximation to posterior distributions arising in location-scale regression models [52,53]; and (f) the prior distribution for set estimation of a multivariate normal mean [11]. Additional applications of (1) can be seen in the numerous books dealing with the Bayesian aspects of multivariate analysis.…”

“…where W p (AE AE AE AE, n) denotes the p-variate Wishart distribution with degrees of freedom n and covariance matrix AE AE AE AE, and if Y has the p-variate normal distribution with zero means and covariance matrix #I p (I p is the pdimensional identity matrix), independent of V, then [4]. This implies that XjV has the p-variate normal distribution with mean vector " " and covariance matrix #V j1 .…”

On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

“…Several authors have derived explicit forms of the compound normal model by mixing the normal with the inverted chi-squared, Downloaded by [The University Of Melbourne Libraries] at 02:39 12 October 2014 the inverse Gaussian or the gamma densities. Among these, we mention Teichroew (1957), Ando and Kaufman (1965), Sankaran (1968), Zellner (1976) and Bhattacharya (1987). Dickey (1967) is among the first to consider the matrix-variate generalization of the multivariate-t distribution where the matrix normal distribution was compounded with the Wishart density to obtain the matrix-T distribution.…”

On maximum information strategies for estimation of stochastic multiple response systems

“…While the traditional multivariate regression model has received most attention in the sampling theory framework, many authors (Zellner 1971, Geisser and Gornfield 1963, Geisser 1965, Ando and Kaufman 1965, Tiao and Zellner 1964 have studied the 'same model from a Bayesian point of view, Consider the model (4) with the further assumption that the disturbances EN are normally distributed with mean zero and variance-covariance matrix QNT; that is, where Strategies for estimation of response systems…”