Analysis often splits change into components. For example, how much of the observed variance is caused by genes or environment? In many cases, the split is ultimately made by the logic of the chain rule, which divides the difference of a product into two terms. Each term quantifies the partial difference associated with change in one component while holding the other component constant. The chain rule is of course widely known. However, this article argues that its deep fundamental role often goes unrecognized. The article shows how simply the basic chain rule unifies Fisher's fundamental theorem of natural selection, the Price equation description of evolutionary change, the Oaxaca-Blinder decomposition of wage differences in economics, the Kitagawa decomposition of mortality differences in demography, many expressions of thermodynamics, and most strikingly back propagation, the core optimization method of modern machine learning and artificial intelligence. The success in creating good designs and finding good solutions in both natural selection and artificial intelligence depends on how the chain rule propagates causes from instances of success or failure back to the underlying genes or parameters of the system. The mathematical analysis presented here shows that, for finite differences, the product rule form of the chain rule yields a basic decomposition of change into two components of a regression equation. That regression decomposition is purely a description of change with no explicit causal meaning. However, simple additional assumptions lead naturally to the modern counterfactual analysis of causality. From that perspective, we can easily understand the causal interpretation that Fisher gave to his fundamental theorem, and we can see the same causal structure in the Oaxaca-Blinder decomposition of economics and in causal analyses across many disciplines.

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