DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

BAYESIAN FORECASTING IN UNIVARIATE
AUTOREGRESSIVE MODELS WITH NORMAL-GAMMA
PRIOR DISTRIBUTION OF UNKNOWN PARAMETERS
IGOR VLADIMIROV AND BEVAN THOMPSON
We consider the problem of computing the mean-square optimal Bayesian predictor in
univariate autoregressive models with Gaussian innovations. The unknown coefficients
of the model are ascribed a normal-gamma prior distribution providing a family of conjugate
priors. The problem is reduced to calculating the state-space realization matrices of
an iterated linear discrete time-invariant system. The system theoretic solution employs a
scalarization technique for computing the power moments of Gaussian random matrices
developed recently by the authors.
Copyright © 2006 I. Vladimirov and B. Thompson. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly
cited.
1. Introduction
We consider the problem of computing the mean-square optimal Bayesian predictor for
a univariate time series whose dynamics is described by an autoregressive model with
Gaussian innovations [3]. The coefficients of the model are assumed unknown and ascribed
a normal-gamma prior distribution [1, page 140] providing a family of conjugate
priors.
The problem reduces to computing the power moments of a square random matrix
which is expressed affinely in terms of a random vector with multivariate Student distribution
[1, page 139]. The latter is a randomized mixture of Gaussian distributions,
thereby allowing us to employ a matrix product scalarization technique developed in
[7] for computing the power moments EX s of Gaussian random matrices X = A + BζC,
where ζ is a standard normal random vector and A, B, C are appropriately dimensioned
constant matrices.
Note that developing an exact algorithm for the power moment problem, alternative
to an approximate solution via Monte Carlo simulation, is complicated by the noncommutativity
of the matrix algebra that is only surmountable in special classes of random
matrices, of which a more general one is treated by sophisticated graph theoretic methods
in [8].
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1099–1108