  We study a distributionally robust optimization formulation (i.e., a min-max
game) for two representative problems in Bayesian nonparametric estimation:
Gaussian process regression and, more generally, linear inverse problems. Our
formulation seeks the best mean-squared error predictor, in an
infinite-dimensional space, against an adversary who chooses the worst-case
model in a Wasserstein ball around a nominal infinite-dimensional Bayesian
model. The transport cost is chosen to control features such as the degree of
roughness of the sample paths that the adversary is allowed to inject. We show
that the game has a well-defined value (i.e., strong duality holds in the sense
that max-min equals min-max) and that there exists a unique Nash equilibrium
which can be computed by a sequence of finite-dimensional approximations.
Crucially, the worst-case distribution is itself Gaussian. We explore
properties of the Nash equilibrium and the effects of hyperparameters through a
set of numerical experiments, demonstrating the versatility of our modeling
framework.
