On the robustness to outliers of the Student-t process

The theory of Bayesian robustness modeling uses heavy-tailed distributions to resolve conflicts of information by rejecting automatically the outlying information in favor of the other sources of information. In particular, the Student's-t process is a natural alternative to the Gaussian process when the data might carry atypical information. Several works attest to the robustness of the Student $t$ process, however, the studies are mostly guided by intuition and focused mostly on the computational aspects rather than the mathematical properties of the involved distributions. This work uses the theory of regular variation to address the robustness of the Student $t$ process in the context of nonlinear regression, that is, the behavior of the posterior distribution in the presence of outliers in the inputs, in the outputs, or in both sources of information. In all these cases, under certain conditions, it is shown that the posterior distribution tends to a quantity that does not depend on the atypical information, then, for every case, the limiting posterior distribution as the outliers tend to infinity is provided. The impact of outliers on the predictive posterior distribution is also addressed. The theory is illustrated with a few simulated examples.