Toward Machine Wald

Abstract

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed by humans because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to think as humans have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables.(2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tends to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with

Multigrid with Rough Coefficients and Multiresolution Operator Decomposition from Hierarchical Information Games

“…Although the problem of finding a fast solver for (1.1) may appear disconnected from that of finding statistical estimators or making decisions from data sampled from an underlying unknown probability distribution, the proposed game theoretic reformulation is, to some degree, analogous to the one developed in Wald's Decision Theory [113], evidently influenced by Von Neumann's Game Theory [111,112] (the generalization of worst case Uncertainty Quantification analysis [83] to sample data/model uncertainty requires an analogous game theoretic formulation [80], see also [79] for how the underlying calculus could be used to guide the discovery of new Selberg identities). We also refer to subsection 1.3 for a review of the correspondence between statistical inference and numerical approximation.…”

A probabilistic model for the numerical solution of initial value problems

“…Chkrebtii et al [7] calls the process of evaluating f (t n , y − tn ) with tentative y − tn to generate z n a model interrogation. From a statistical perspective, this concept of active model interrogation is similar to the sequential analysis of Wald [63,43].…”

A modern retrospective on probabilistic numerics

“…In the statistical literature, there is also a long history of Bayesian optimal experimental design, in parametric and non-parametric contexts [Lindley, 1956, Piiroinen, 2005. The extent to which these principles can be used to design optimal numerical methods automatically (rather than by inspired guesswork on the mathematician's part, à la Larkin) remains a major open question, analogous to the automation of statistical reasoning envisioned by Wald and subsequent commentators on his work [Owhadi and Scovel, 2017b].…”