Towards 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 \emph{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 \emph{think} as \emph{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, tend 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 Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.

37 pages

References in corpus (6)

• On Hoeffding's inequalities
• Nonparametric Bernstein-von Mises theorems in Gaussian white noise
• On the Brittleness of Bayesian Inference
• Brittleness of Bayesian Inference Under Finite Information in a Continuous World
• Exact inequalities for sums of asymmetric random variables, with applications
• Optimal uncertainty quantification for legacy data observations of Lipschitz functions

Cited by in corpus (4)

• Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients
• Universal Scalable Robust Solvers from Computational Information Games and fast eigenspace adapted Multiresolution Analysis
• On testing the simulation theory
• Decision Theoretic Bootstrapping