Towards a Unified Framework for Statistical and Mathematical Modeling

TL;DR

This paper proposes a unified framework connecting statistical and mathematical modeling using the concept of identification, facilitating better interpretation, comparison, and integration of models across scientific disciplines.

Contribution

It introduces a shared language based on identification and bounds, extending causal inference methods to both statistical and mathematical models.

Findings

Develops a formal connection between statistical and mathematical models.

Uses bounds to analyze model plausibility and comparison.

Illustrates the framework with a pharmacodynamic model for hypertension.

Abstract

Within the biological, physical, and social sciences, there are two broad quantitative traditions: statistical and mathematical modeling. Both traditions have the common pursuit of advancing our scientific knowledge, but these traditions have developed largely independently using distinct languages and inferential frameworks. This paper uses the notion of identification from causal inference, a field originating from the statistical modeling tradition, to develop a shared language. I first review foundational identification results for statistical models and then extend these ideas to mathematical models. Central to this framework is the use of bounds, ranges of plausible numerical values, to analyze both statistical and mathematical models. I discuss the implications of this perspective for the interpretation, comparison, and integration of different modeling approaches, and illustrate the framework with a simple pharmacodynamic model for hypertension. To conclude, I describe areas where the approach taken here should be extended in the future. By formalizing connections between statistical and mathematical modeling, this work contributes to a shared framework for quantitative science. My hope is that this work will advance interactions between these two traditions.