Causal attribution by the chain rule: unifying natural selection, learning, economics, and other disciplines

1. Problem Statement

Scientific disciplines frequently attempt to decompose observed changes in a system into distinct causal components. Examples include:

• Evolutionary Biology: Separating the change in mean fitness caused by natural selection (gene frequency changes) from changes caused by environmental shifts or genetic context.
• Economics: Decomposing wage gaps between groups (e.g., gender or race) into differences attributable to observable characteristics (education) versus differences in the returns to those characteristics (potential discrimination).
• Machine Learning: Attributing changes in model performance to specific parameter updates via backpropagation.

Despite the apparent diversity of these fields, the paper argues that the underlying mathematical logic for these decompositions is often unrecognized. The central problem is to identify the unifying algebraic structure that allows for the attribution of cause across these disparate domains and to clarify how this structure transitions from a purely descriptive mathematical identity to a framework for counterfactual causal analysis.

2. Methodology

The paper employs a rigorous mathematical derivation based on finite differences and the product rule (a specific application of the chain rule).

• The Product Rule for Finite Differences:
  The author starts with the change in a product of two variables, $z = bx$. The finite difference $\Delta z = z' - z$ is expanded as:
  $$\Delta z = b\Delta x + x'\Delta b$$
  Here, the first term represents the change in $x$ holding $b$ constant, and the second term represents the change in $b$ weighted by the new value of $x$ ($x'$).

• Extension to Vectors and Regression:
  The method generalizes this to vector dot products, representing a regression model where a focal variable $z$ is a linear combination of predictors $x$ weighted by coefficients $b$ (plus an error term $\epsilon$).
  For a population mean $\bar{z} = \mathbf{b} \cdot \bar{\mathbf{x}}$, the change in the mean is derived as:
  $$\Delta \bar{z} = \mathbf{b} \cdot \Delta \bar{\mathbf{x}} + \bar{\mathbf{x}}' \cdot \Delta \mathbf{b}$$
  This equation partitions the total change into two components:

  1. Direct Effect: Change in the predictors ($\Delta \bar{\mathbf{x}}$) holding coefficients ($\mathbf{b}$) constant.
  2. Contextual/Interaction Effect: Change in the coefficients ($\Delta \mathbf{b}$) weighted by the new predictor values ($\bar{\mathbf{x}}'$).

• Counterfactual Interpretation:
  The paper argues that while the equation itself is purely algebraic (descriptive), it naturally leads to counterfactual causal analysis when one interprets the terms as "what-if" scenarios. By holding one set of factors constant (the "counterfactual" condition), one can isolate the causal contribution of the changing factors.

3. Key Contributions

The paper makes three primary contributions:

1. Mathematical Unification: It demonstrates that Fisher's Fundamental Theorem of Natural Selection, the Price Equation, the Oaxaca-Blinder decomposition (economics), the Kitagawa decomposition (demography), and thermodynamic decompositions all reduce to the same product rule for finite differences applied to a regression equation.
2. Causal Derivation: It shows how modern counterfactual causal analysis arises naturally from the chain rule. The act of "holding one factor constant while varying another" in the product rule is mathematically identical to the logic used in causal inference to isolate specific forces.
3. Biological and AI Synthesis: It unifies natural selection and artificial intelligence (specifically backpropagation). Both processes rely on attributing "credit" (or error) to underlying parameters (genes or weights) by propagating changes through a chain-rule-like calculation.

4. Results and Applications

A. Fisher's Fundamental Theorem of Natural Selection

The paper re-derives Fisher's theorem using the finite difference product rule.

• Context: $z$ is fitness ($w$), $x$ are gene frequencies ($p$), and $b$ are average effects (partial regression coefficients).
• Decomposition: $\Delta \bar{w} = \mathbf{b} \cdot \Delta \mathbf{p} + \mathbf{p}' \cdot \Delta \mathbf{b}$.
• Interpretation:
  • The first term ($\mathbf{b} \cdot \Delta \mathbf{p}$) represents the change in mean fitness due solely to natural selection acting on gene frequencies, assuming the average effects of genes remain constant. This is identified as the additive genetic variance.
  • The second term captures changes due to the environment, genetic interactions (epistasis/dominance), and changes in the mapping from genotype to fitness.
• Insight: Fisher's "theorem" is not a claim about the total evolution of fitness, but a specific causal attribution isolating the direct force of selection from the complex context.

B. Oaxaca-Blinder Decomposition (Economics)

• Context: Decomposing wage gaps between two groups.
• Decomposition: The same equation applies, where $\Delta \bar{x}$ is the difference in characteristics (e.g., education) and $\Delta \mathbf{b}$ is the difference in returns (coefficients).
• Contrast with Fisher: While Fisher focused on the first term (selection), economists often focus on the second term ($\Delta \mathbf{b}$) to detect discrimination (unexplained wage gaps). The math is identical; the causal question differs.

C. Backpropagation and Machine Learning

• The paper posits that backpropagation in neural networks is an application of this same chain-rule logic.
• In both natural selection and AI, the system identifies "good designs" by attributing success or failure back to specific parameters (genes or weights).
• Both systems face challenges with nonlinear interactions, but the partial derivative (or finite difference) approach allows for effective optimization by updating parameters based on partial perturbations.

D. Other Disciplines

• Demography (Kitagawa): Separates mortality rate differences into changes in age structure vs. changes in age-specific mortality rates.
• Thermodynamics: Decomposes changes in ensemble means into changes in probability distributions vs. changes in observable values.

5. Significance

• Conceptual Clarity: The paper demystifies complex causal analyses in diverse fields by revealing they all rest on a single, elementary algebraic identity: the product rule for finite differences.
• Resolution of Disagreements: It explains why debates persist in fields like evolutionary biology (e.g., regarding Fisher's theorem). Disagreements often stem not from mathematical errors, but from different choices regarding which counterfactual condition to hold constant (i.e., which term in the decomposition is treated as the "cause" of interest).
• Interdisciplinary Bridge: It creates a formal bridge between evolutionary biology, econometrics, demography, and machine learning, suggesting that the logic of "learning" and "selection" is fundamentally the same process of causal attribution via the chain rule.
• Causal Framework: It provides a simplified pathway to counterfactual analysis, showing that the complex machinery of modern causal inference (e.g., Pearl's do-operators) is deeply rooted in the simple logic of holding variables constant to isolate effects.

In conclusion, Frank argues that the "miracles" of natural selection and artificial intelligence are grounded in the simple mathematical ability to decompose change into partial components, a process governed by the chain rule. Recognizing this unity clarifies the causal logic underlying these systems.