  This paper considers the robust and efficient implementation of Gaussian
process regression with a Student-t observation model. The challenge with the
Student-t model is the analytically intractable inference which is why several
approximative methods have been proposed. The expectation propagation (EP) has
been found to be a very accurate method in many empirical studies but the
convergence of the EP is known to be problematic with models containing
non-log-concave site functions such as the Student-t distribution. In this
paper we illustrate the situations where the standard EP fails to converge and
review different modifications and alternative algorithms for improving the
convergence. We demonstrate that convergence problems may occur during the
type-II maximum a posteriori (MAP) estimation of the hyperparameters and show
that the standard EP may not converge in the MAP values in some difficult
cases. We present a robust implementation which relies primarily on parallel EP
updates and utilizes a moment-matching-based double-loop algorithm with
adaptively selected step size in difficult cases. The predictive performance of
the EP is compared to the Laplace, variational Bayes, and Markov chain Monte
Carlo approximations.
