Semiparametric Efficiency Theory as Differential Calculus on a Space of Probability Distributions

Razieh Nabi

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arXiv · 2606.22784

Semiparametric Efficiency Theory as Differential Calculus on a Space of Probability Distributions

Razieh Nabi

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Abstract

Semiparametric efficiency theory provides the mathematical foundation for influence-function-based estimation, including one-step estimators, targeted minimum loss estimators, and many modern inferential methods used in causal inference and missing data analysis. Despite its widespread use, the theory is often presented through a collection of technical constructions whose geometric meaning remains opaque. As a result, influence functions are often derived and applied without an intuitive understanding of the principles connecting scores, tangent spaces, nuisance tangent spaces, and efficient influence functions. This tutorial develops a geometric exposition of semiparametric efficiency theory as a form of differential calculus on a space of probability distributions. Drawing systematic parallels with ordinary multivariable calculus, we show that paths of distributions play the role of curves, scores play the role of velocity vectors, influence functions play the role of gradients, and efficient influence functions arise as projected gradients. This perspective provides a unified explanation for several foundational questions, including why perturbation directions are represented by functions, why tangent spaces depend only on the statistical model whereas nuisance tangent spaces depend on the parameter of interest, and why efficient influence functions arise through orthogonal projection. The resulting framework offers a geometric perspective on semiparametric efficiency theory and influence-function-based inference.

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Razieh Nabi. 2026-06-22. Semiparametric Efficiency Theory as Differential Calculus on a Space of Probability Distributions. https://arxiv.org/abs/2606.22784

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