Three Multidimensional-integral Identities with Bayesian Applications

Abstract

The first identity (Section 2) expresses a moment of a product of multivariate $t$ densities as an integral of dimension one less than the number of factors. This identity is applied to inference concerning the location parameters of a multivariate normal distribution. The second identity (Section 3) expresses the density of a linear combination of independently distributed multivariate $t$ vectors, a multivariate Behrens-Fisher density (Cornish (1965)), as an integral of dimension one less than the number of summands. The two-summand version of the second identity is essentially equivalent to the two-factor version of the first identity. A synthetic representation is given for the random vector, generalizing Ruben's (1960) representation in the univariate case. The second identity is applied to multivariate Behrens-Fisher problems. The third identity (Section 4), due to Picard (Appell and Kampe de Feriet (1926)), expresses the moments of Savage's (1966) generalization of the Dirichlet distribution as a one-dimensional integral. A generalization of Picard's identity is given. Picard's identity is applied to inference about multinomial cell probabilities, to components-of-variance problems, and to inference from a likelihood function under a Student $t$ distribution of errors.