  We introduce a distributionally robust minimium mean square error estimation
model with a Wasserstein ambiguity set to recover an unknown signal from a
noisy observation. The proposed model can be viewed as a zero-sum game between
a statistician choosing an estimator -- that is, a measurable function of the
observation -- and a fictitious adversary choosing a prior -- that is, a pair
of signal and noise distributions ranging over independent Wasserstein balls --
with the goal to minimize and maximize the expected squared estimation error,
respectively. We show that if the Wasserstein balls are centered at normal
distributions, then the zero-sum game admits a Nash equilibrium, where the
players' optimal strategies are given by an {\em affine} estimator and a {\em
normal} prior, respectively. We further prove that this Nash equilibrium can be
computed by solving a tractable convex program. Finally, we develop a
Frank-Wolfe algorithm that can solve this convex program orders of magnitude
faster than state-of-the-art general purpose solvers. We show that this
algorithm enjoys a linear convergence rate and that its direction-finding
subproblems can be solved in quasi-closed form.
