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The nature of mathematical models

Andrea De Gaetano

TL;DR

The paper defines a rigorous, Hilbert-space–based foundation for mathematical modeling by treating a model as an operator on $\mathcal{H}=L^2(\Omega,\mathcal{F},P)$ with a model manifold $\mathcal{M}_{\boldsymbol{X}}$ and a prediction surface $\mathcal{M}_{\boldsymbol{x}}$ that inhabit $\mathcal{H}$ and $\mathbb{R}^n$ respectively. It develops a geometry-driven link between abstract modeling in $\mathcal{H}$ and concrete estimation in case space $\mathbb{R}^n$, using projections for linear cases and tangent-space (affine) linearizations for nonlinear cases, with Jacobians playing a central role. The work extends classical projection results to affine subspaces and provides a reparameterization approach to define local tangent spaces, connecting conditional estimation and information geometry concepts while clarifying how to map between abstract model structure and computational procedures. These contributions yield a principled framework for model identification and parameter estimation across disciplines, offering a precise language for the relationship between theoretical model construction and practical statistical inference.

Abstract

Modeling has become a widespread, useful tool in mathematics applied to diverse fields, from physics to economics to biomedicine. Practitioners of modeling may use algebraic or differential equations, to the elements of which they attribute an intuitive relationship with some relevant aspect of reality they wish to represent. More sophisticated expressions may include stochasticity, either as observation error or system noise. However, a clear, unambiguous mathematical definition of what a model is and of what is the relationship between the model and the real-life phenomena it purports to represent has so far not been formulated. The present work aims to fill this gap, motivating the definition of a mathematical model as an operator on a Hilbert space of random variables, identifying the experimental realization as the map between the theoretical space of model construction and the computational space of statistical model identification, and tracing the relationship of the geometry of the model manifold in the abstract setting with the corresponding geometry of the prediction surfaces in statistical estimation.

The nature of mathematical models

TL;DR

The paper defines a rigorous, Hilbert-space–based foundation for mathematical modeling by treating a model as an operator on with a model manifold and a prediction surface that inhabit and respectively. It develops a geometry-driven link between abstract modeling in and concrete estimation in case space , using projections for linear cases and tangent-space (affine) linearizations for nonlinear cases, with Jacobians playing a central role. The work extends classical projection results to affine subspaces and provides a reparameterization approach to define local tangent spaces, connecting conditional estimation and information geometry concepts while clarifying how to map between abstract model structure and computational procedures. These contributions yield a principled framework for model identification and parameter estimation across disciplines, offering a precise language for the relationship between theoretical model construction and practical statistical inference.

Abstract

Modeling has become a widespread, useful tool in mathematics applied to diverse fields, from physics to economics to biomedicine. Practitioners of modeling may use algebraic or differential equations, to the elements of which they attribute an intuitive relationship with some relevant aspect of reality they wish to represent. More sophisticated expressions may include stochasticity, either as observation error or system noise. However, a clear, unambiguous mathematical definition of what a model is and of what is the relationship between the model and the real-life phenomena it purports to represent has so far not been formulated. The present work aims to fill this gap, motivating the definition of a mathematical model as an operator on a Hilbert space of random variables, identifying the experimental realization as the map between the theoretical space of model construction and the computational space of statistical model identification, and tracing the relationship of the geometry of the model manifold in the abstract setting with the corresponding geometry of the prediction surfaces in statistical estimation.

Paper Structure

This paper contains 24 sections, 8 theorems, 47 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Notation
  4. Standard notation
  5. Further notation
  6. Geometry of Estimation in Case Space ($\mathbb{R}^n$)
  7. Linear relationship in case space: geometry
  8. Linear relationship in case space: estimation by projection
  9. Nonlinear relationship in case space
  10. Geometry of Estimation in the Hilbert space $\mathcal{H}$ of random variables
  11. Preliminaries
  12. The nature of mathematical models: definitions
  13. Linear and nonlinear models
  14. Mapping $\mathcal{H}$ to $\mathbb{R}^n$
  15. Summary
  16. ...and 9 more sections

Key Result

Theorem 1

Let $(\Omega, \mathcal{F},P)$ be a probability space, $(\mathbb{R}^n,\mathcal{B}_n)$ and $(\mathbb{R}^m,\mathcal{B}_m)$ measurable spaces, $U:\Omega\rightarrow\mathbb{R}^n$ a random variable. Then a function $X:\Omega\rightarrow\mathbb{R}^m$ is $\sigma(U)$-$\mathcal{B}_m$-measurable iff $\,\, \exist

Figures (6)

  • Figure 1: The geometry of a linear relationship in case space
  • Figure 2: Nonlinear relationship in case space: original geometry
  • Figure 3: By the Doob-Dynkin lemma this diagram commutes for measurable functions
  • Figure 4: Sampling as a map from $\mathcal{H}$ to $\mathbb{R}^n$.
  • Figure 5: Nonlinear relationship in case space: reparametrized geometry
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem : Doob-Dynkin
  • Theorem : Canonical norm
  • Definition
  • Definition
  • Theorem
  • Theorem : Hilbert projection theorem
  • Proposition
  • Proposition
  • Definition
  • Remark
  • ...and 7 more