This chapter reviews the correspondence between approximation theory and statistical inference.

The introduction reviews, summarizes, and illustrates fundamental connections among Bayesian inference, numerical quadrature, Gausssian process regression, polyharmonic splines, information-based complexity, optimal recovery, and game theory that form the basis for the book. This is followed by describing a sample of the results derived from these interplays; including those in numerical homogenization, operator-adapted wavelets, fast solvers, and Gaussian process regression. It finishes with an outline of the structure of the book.

This chapter generalizes and extends the treatment of optimal recovery splines from Sobolev spaces to Banach spaces equipped with quadratic norm and nonstandard dual pairing.

This chapter introduces optimal recovery games on Banach spaces, presents their natural lift to mixed strategies, and then characterizes their saddle points interms of Gaussian measures, cylinder measures, and fields. The canonical Gaussian field is shown to be a universal field in the sense that its conditioningwith respect to linear measurements producesoptimal strategies. When those measurements form a nested hierarchy, hierarchies of optimal approximations form a martingale obtained by conditioningthe Gaussian field on the filtration formed by those measurements.

This chapter presents the theory of optimal recovery in the setting of Sobolev spaces andthe context of information-based complexity. It also describes optimal recovery splines, their variational properties, and their minmax optimality characterization.