The Nature of Mathematical Models

Mathematical modeling has become pervasive in applications, not only in physics or economics, but also in biomedicine and other “soft” sciences. To the conceptual formulation of a model, there often follows its identification by statistical parameter estimation, given available observations. While the nature of the modeling process as well as its relationship with the attending statistical computations could both appear obvious to the practitioner, it may be useful to formalize them in a precise way. Insight into the process of (linear and nonlinear) model parameter estimation can be obtained from the description of the geometry of estimation in case space. The objective then is to describe the geometry of modeling in the abstract, and to show how the correspondence between the conceptual context of the model as an operator in the Hilbert space of finite-variance random variables and the computational context in ${\mathbb{R}}^n$ can be formally represented. This work formalizes the geometric correspondence between model manifolds in the Hilbert space of random variables and the geometry of statistical estimation in case space, integrating classical tools (Hilbert spaces, manifolds, projections) into a unified framework for understanding modeling and estimation.