  This paper considers the Laplace method to derive approximate inference for
the Gaussian process (GP) regression in the location and scale parameters of
the Student-t probabilistic model. This allows both mean and variance of the
data to vary as a function of covariates with the attractive feature that the
Student-t model has been widely used as a useful tool for robustifying data
analysis. The challenge in the approximate inference for the GP regression with
the Student-t probabilistic model, lies in the analytical intractability of the
posterior distribution and the lack of concavity of the log-likelihood
function. We present the natural gradient adaptation for the estimation process
which primarily relies on the property that the Student-t model naturally has
orthogonal parametrization with respect to the location and scale paramaters.
Due to this particular property of the model, we also introduce an alternative
Laplace approximation by using the Fisher information matrix in place of the
Hessian matrix of the negative log-likelihood function. According to
experiments this alternative approximation provides very similar posterior
approximations and predictive performance when compared to the traditional
Laplace approximation. We also compare both of these Laplace approximations
with the Monte Carlo Markov Chain (MCMC) method. Moreover, we compare our
heteroscedastic Student-t model and the GP regression with the heteroscedastic
Gaussian model. We also discuss how our approach can improve the inference
algorithm in cases where the probabilistic model assumed for the data is not
log-concave.
