Bringing errors into focus

TL;DR

This paper explores recent theoretical advances in error propagation, linking differential calculus, differential geometry, and Dirichlet forms to improve understanding of errors in complex models, including infinite-dimensional spaces.

Contribution

It introduces a unified framework connecting error propagation, differential geometry, and Dirichlet forms, extending to infinite-dimensional models and linking to statistical concepts.

Findings

Error propagation can be analyzed with first or second order calculus.

Dirichlet forms provide a versatile framework for infinite-dimensional models.

The square of the Dirichlet form's field operator relates to Fisher information.

Abstract

This lecture presents recent advances in the theory of errors propagation. We first explain in which cases the propagation of errors may be performed with a first order differential calculus or needs a second order differential calculus. Then we point out the link between error propagation and the concept of second order vector in differential geometry, emphasizing the existence of a slight ambiguity concerning the bias operator. The third part in devoted to the powerful framework of Dirichlet forms whose main feature is to apply easily to infinite dimensional models including the Wiener space (giving an interpretation of Malliavin calculus in terms of errors), the Poisson space and the Monte Carlo space. In the fourth part we show how an error in the usual mathematical sense, i.e. an approximate quantity, may yield a Dirichlet form and we introduce the four bias operators. Eventually we connect the Dirichlet form with statistics by identifying the square of field operator with the inverse of the Fisher information matrix.