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Towards a Unified Framework for Statistical and Mathematical Modeling

This paper proposes a unified framework for statistical and mathematical modeling by leveraging causal inference concepts, particularly identification and bounds, to create a shared language that facilitates the interpretation, comparison, and integration of these traditionally distinct quantitative traditions.

Original authors: Paul N Zivich

Published 2026-03-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Paul N Zivich

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to figure out how much a specific medicine (let's call it "Super-Drug") lowers blood pressure. In the world of science, there are two main groups of people trying to solve this puzzle, and they speak very different languages.

The Two Tribes:

The Statisticians (The Observers): These folks look at real-world data. They say, "Let's watch 1,000 people, give half the drug and half a placebo, and see what happens." They rely on patterns, surveys, and randomized trials. Their language is all about probabilities and "what we saw."
The Mathematicians (The Architects): These folks build theoretical machines. They say, "Let's write a set of equations describing how the drug moves through the body, how it hits the immune system, and how that lowers pressure." They rely on logic, formulas, and "what should happen" based on theory. They don't always need to watch people first; they just need a good theory.

The Problem:
For a long time, these two groups have been working in separate rooms. The Statisticians think the Mathematicians are too theoretical and might be guessing. The Mathematicians think the Statisticians are just looking at noise and missing the big picture. They struggle to compare their results or combine their strengths because they don't have a shared dictionary.

The Solution: A Shared Language of "Bounds"
The author of this paper, Paul Zivich, wants to build a bridge. He uses a concept from the Statisticians called "Identification" and applies it to the Mathematicians.

Think of "Identification" like trying to find a lost key in a dark room.

Point Identification: You turn on the light, and you see the key is exactly under the chair. You know the answer perfectly.
Partial Identification (The "Bounds"): You can't turn on the light. But you know the key is somewhere between the door and the window. You don't know the exact spot, but you know it's not in the kitchen. You have a range (or "bound") of where the answer could be.

The Big Idea:
Zivich argues that both Statisticians and Mathematicians should stop trying to find the exact answer immediately and instead focus on calculating these Bounds (the safe range of possibilities).

For Statisticians: They usually try to find the exact answer by assuming their data is perfect. Zivich says, "No, let's admit we don't know everything. Let's calculate the widest possible range of answers that makes sense given our data."
For Mathematicians: They usually build a perfect machine and plug in numbers. Zivich says, "Your machine is only as good as the numbers you put in. If you aren't sure about those numbers, don't just give us one answer. Give us a range of answers based on the uncertainty of your inputs."

The "Vacuous" vs. "Non-Vacuous" Model (The Empty Box Analogy)
The paper introduces a cool distinction:

A "Vacuous" Model: Imagine you have a box that can hold anything from a feather to a boulder. If you ask, "What's inside?" and the box can hold anything, the box tells you nothing new. It's "vacuous" (empty of information). In math terms, if your equations are so flexible they can produce any result, they aren't actually helping you understand the world.
A "Non-Vacuous" Model: This is a box that can only hold things between 5 and 10 pounds. If you ask, "What's inside?" and the box says "It's between 5 and 10 pounds," that's useful! It has ruled out the feather and the boulder.

Zivich wants Mathematicians to build "Non-Vacuous" models. They should use their theories to rule out impossible answers, narrowing the "bounds" even before they look at real data.

The Case Study: The Blood Pressure Drug
To prove this works, Zivich builds a simple model for a blood pressure drug (amlodipine).

He sets up the math equations (the "Architect" part).
Instead of guessing the exact numbers for how the drug works, he admits he doesn't know them perfectly.
He uses outside data (like studies on body weight or drug concentration) to create ranges for those numbers.
He runs the math with the "best case" and "worst case" numbers from those ranges.
The Result: Instead of saying "The drug lowers pressure by 12.4 mmHg," he says, "Based on our math and the data we have, the drug definitely lowers pressure by at least 0.23 mmHg and at most 0.91 mmHg."

Why This Matters
This approach is like a safety net.

It forces scientists to be honest about what they don't know.
It allows the "Observers" and "Architects" to talk to each other. The Architect can say, "My theory says the answer must be between X and Y." The Observer can say, "My data says the answer is between A and B." If the ranges overlap, they are on the same page!
It helps us spot bad science. If a model claims to know the exact answer but its "bounds" are actually huge, we know the model is shaky.

In a Nutshell:
This paper is a call to stop fighting over who has the "true" answer. Instead, let's all agree to calculate the safe range of possibilities. By using a shared language of "bounds," Statisticians and Mathematicians can finally work together to build a clearer, more honest picture of how the world works.

1. Problem Statement

The paper addresses the historical and methodological divide between two dominant quantitative traditions in the biological, physical, and social sciences:

Statistical (Empirical/Phenomenological) Modeling: Relies on observed data to estimate parameters (e.g., regression, g-formula, randomized trials).
Mathematical (Mechanistic) Modeling: Relies on theoretical representations and formulas (e.g., differential equations, agent-based models) without necessarily requiring direct observation for structure.

While both aim to advance scientific knowledge, they operate with distinct languages and inferential frameworks. There is a lack of a shared language to formally compare, integrate, or evaluate these approaches. Specifically, the concept of identification (determining if a causal parameter can be uniquely derived from data) is well-defined in statistical causal inference but lacks a rigorous parallel in mathematical modeling, where "identifiability" often refers to parameter estimation from data rather than the derivation of causal effects from theory.

2. Methodology

The author proposes a unified framework based on causal inference, specifically utilizing the concept of bounds (ranges of plausible numerical values) to analyze both statistical and mathematical models.

A. Conceptual Foundation: The Interest Parameter

The framework centers on an "interest parameter" ( $\psi$ ), such as the Average Causal Effect (ACE). The goal is to express $\psi$ as a function of available information (data or theory).

Notation: Uses potential outcomes ( $Y^a$ ) to define causal effects.
Identification: Defined as expressing $\psi$ $ψ$ as a function of available information.
- Point Identification: $\psi$ maps to a single unique value.
- Partial Identification: $\psi$ maps to a range (bounds) within the parameter space $\Psi$ .

B. Statistical Modeling Review

The paper reviews how statistical models handle identification:

Partial Identification: Without assumptions, bounds for $\psi$ are wide (e.g., $[-1, 1]$ for binary outcomes).
Point Identification: Achieved via assumptions like randomization (marginal exchangeability) or no unmeasured confounding (conditional exchangeability), leading to the g-formula.
Estimation: Requires an additional assumption of correct model specification (the statistical model contains the true data-generating process).

C. Mathematical Modeling Extension

The core innovation is extending identification to mathematical models where parameters are not necessarily estimated from the specific dataset but derived from theory.

Model Structure: A mathematical model is defined by functions (e.g., $g$ for immune response, $h$ for infection status) and parameters ( $\theta, \lambda$ ).
Vacuous vs. Non-Vacuous Models:
- Vacuous Model: The functions and parameter spaces are so flexible that the model can produce any value in the theoretical parameter space $\Psi$ . It provides no information beyond the axioms of probability.
- Non-Vacuous Model: Structural assumptions restrict the possible values of $\psi$ , ruling out certain outcomes before parameters are even chosen.
The "Model-Capture" Assumption: To derive bounds for a mathematical model, one must assume that the true parameters ( $\theta^*, \lambda^*$ $θ^{*}, λ^{*}$ ) lie within a specific subset of the parameter space ( $\Theta^*, \Lambda^*$ $Θ^{*}, Λ^{*}$ ) justified by external evidence (e.g., pharmacokinetic studies, animal models).
- Under this assumption, $\psi$ is partially identified within the bounds:
  $\psi \in \left[ \min_{\theta, \lambda \in \Theta^* \times \Lambda^*} \bar{\psi}(\theta, \lambda), \max_{\theta, \lambda \in \Theta^* \times \Lambda^*} \bar{\psi}(\theta, \lambda) \right]$

3. Key Contributions

Unified Language of Bounds: The paper establishes "bounds" as the common metric for both traditions. Statistical models narrow bounds via data assumptions; mathematical models narrow bounds via structural constraints and parameter ranges derived from external theory.
Formalization of Mathematical Identification: It redefines identification for mathematical models not as parameter estimation, but as the derivation of causal effect bounds based on the "model-capture" assumption.
Distinction between Structure and Parameters: By introducing the concept of vacuous models, the author separates structural assumptions (encoded in function choice) from parameter uncertainty. This clarifies when a model is truly informative versus when it is merely a flexible mathematical container.
Reframing Exchangeability: The paper argues that exchangeability in mathematical modeling applies to the external information used to justify parameter ranges (model-capture), not necessarily the treatment assignment itself (which is often set by the modeler).

4. Results: Case Study (Amlodipine and Hypertension)

The framework is applied to a pharmacodynamic model for the effect of amlodipine (a calcium channel blocker) on systolic blood pressure.

Setup: The model links treatment ( $A$ ) to effective drug concentration ( $M$ ) to blood pressure outcome ( $Y$ ).
Process:
1. The model was initially shown to be vacuous (capable of producing any ACE).
2. Constraints were applied:
  - $f(b)$ : Baseline blood pressure distribution derived from NHANES data.
  - $\Theta^*$ : Drug concentration parameters constrained by pharmacokinetic data (South Korean study).
  - $\Lambda^*$ : Pharmacodynamic parameters constrained by clinical trial data.
3. Calculation: The minimum and maximum ACE were computed across the constrained parameter space.
Outcome: The model produced partial identification bounds of $\Psi^* = [0.23, 0.91]$ . This indicates a substantial protective effect of amlodipine, but the width of the interval reflects the uncertainty in the external data sources used to constrain the parameters.
Evaluation: The framework allowed the author to critique the model's validity, identifying issues such as:
- Mismatch between the source population of the parameter data (South Korean males) and the target population (US adults).
- Lack of uncertainty propagation from the statistical data (NHANES) into the mathematical model.
- Potential oversimplification of physiological mechanisms (e.g., ignoring body weight differences).

5. Significance and Future Directions

Bridging the Gap: The paper provides a rigorous method to compare statistical and mathematical models, allowing researchers to see how structural assumptions in mechanistic models interact with data-driven assumptions in statistical models.
Model Evaluation: It offers a systematic way to evaluate mathematical models by explicitly stating the "model-capture" assumptions and the resulting bounds, rather than relying solely on point estimates or goodness-of-fit metrics.
Future Extensions: The author identifies several areas for future work:
- Moving from bounds to distributions by incorporating "belief" or uncertainty about parameter choices.
- Adapting the framework for interference (where one unit's treatment affects another).
- Integrating differential equations into the identification framework.
- Formalizing calibration procedures in mathematical modeling using identification theory.

In conclusion, Zivich's work argues that by treating mathematical models through the lens of causal identification and bounds, the scientific community can better assess the validity, limitations, and utility of mechanistic models, fostering a more integrated approach to quantitative science.