  Gaussian process model for vector-valued function has been shown to be useful
for multi-output prediction. The existing method for this model is to
re-formulate the matrix-variate Gaussian distribution as a multivariate normal
distribution. Although it is effective in many cases, re-formulation is not
always workable and is difficult to apply to other distributions because not
all matrix-variate distributions can be transformed to respective multivariate
distributions, such as the case for matrix-variate Student$-t$ distribution. In
this paper, we propose a unified framework which is used not only to introduce
a novel multivariate Student$-t$ process regression model (MV-TPR) for
multi-output prediction, but also to reformulate the multivariate Gaussian
process regression (MV-GPR) that overcomes some limitations of the existing
methods. Both MV-GPR and MV-TPR have closed-form expressions for the marginal
likelihoods and predictive distributions under this unified framework and thus
can adopt the same optimization approaches as used in the conventional GPR. The
usefulness of the proposed methods is illustrated through several simulated and
real data examples. In particular, we verify empirically that MV-TPR has
superiority for the datasets considered, including air quality prediction and
bike rent prediction. At last, the proposed methods are shown to produce
profitable investment strategies in the stock markets.
