resulting method could lack robustness to rough coefficients, even under the assumption that coefficients are stationary [@BalJing10]). Furthermore, to envisage applications to multigrid methods, the computation of these basis functions must also be provably localized [@BaLip10; @OwZh:2011; @MaPe:2012; @GrGrSa2012; @OwhadiZhangBerlyand:2014; @HouLiu2015] and compatible with nesting strategies [@OwhadiZhangBerlyand:2014]. In [@Owhadi:2014], it has been shown that this process of identification (of accurate basis elements for numerical homogenization), could, in principle, be guided through its reformulation as a Bayesian Inference problem in which the source term $g$ in is replaced by noise $\xi$ and one tries to estimate the value of the solution at a given point based on a finite number of observations. In particular it was found that Rough Polyharmonic Splines [@OwhadiZhangBerlyand:2014] and Polyharmonic Splines [@Harder:1972; @Duchon:1976; @Duchon:1977; @Duchon:1978] can be re-discovered as solutions of Gaussian filtering problems. This paper is inspired by the suggestion that this link between numerical homogenization and Bayesian Inference (and the link between Numerical Quadrature and Bayesian Inference [@Poincare:1896; @Diaconis:1988; @Shaw:1988; @Hagan:1991; @Hagan:1992]) are not coincidences but particular instances of mixed strategies for underlying information games and that optimal or near optimal methods could be obtained by identifying such games and their optimal strategies.

The process 