Resolution, Nonlinearity, Constraints, Confidence, Design, Minimax and Bayes

Of those things that can be estimated well in an inverse problem, which are best to estimate? Backus-Gilbert resolution theory answers a version of this question for linear (or linearized) inverse problems in Hilbert spaces with additive zero-mean errors with known, finite covariance, and no constraints on the unknown other than the data. This paper extends Backus-Gilbert resolution: it defines the resolution and Bayes resolution of an estimator, intrinsic minimax and Bayes resolution, and intrinsic minimax and Bayes design resolution. Intrinsic resolution is the smallest value of a penalty among functionals that can be estimated with controlled (minimax or Bayes) risk. Intrinsic minimax resolution includes Backus-Gilbert resolution and subtractive optimally localized averages (SOLA) as special cases. Intrinsic design resolution is the smallest value of a penalty among functionals that can be estimated with controlled (minimax or Bayes) risk using observations with a controlled acquisition cost. Intrinsic resolution wraps the classical problem of choosing an optimal estimator of an abstract parameter inside the problem of choosing an optimal parameter to estimate. Intrinsic design resolution adds another layer: optimizing what to observe. The definitions apply to inverse problems with constraints, to nonlinear inverse problems, to nonlinear and biased estimators, to general measures of risk (not just the variance of unbiased estimators), and to abstract penalties not necessarily related to “spread.” For illustration, the resolution of “strict bounds” confidence intervals is defined.