Bringing errors into focus
Provides a theoretical framework for error propagation in stochastic models, but is primarily a conceptual synthesis rather than a practical breakthrough.
This lecture advances the theory of error propagation by linking it to differential geometry and Dirichlet forms, showing that second-order calculus is needed in certain cases and that Dirichlet forms unify error analysis across infinite-dimensional models including Wiener and Poisson spaces.
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.