Efficient influence functions connect causal identification to semiparametric estimation. They provide the first-order derivative of a target functional, the efficiency bound, an estimating equation, and the correction used by one-step and targeted estimators.
Regular parametric submodels
Definition. A regular parametric submodel \(\{P_\varepsilon:|\varepsilon|<\delta\}\) through \(P=P_0\) has score \[ s(O)= \left.\frac{\partial}{\partial\varepsilon} \log p_\varepsilon(O)\right|_{\varepsilon=0}, \qquad Ps=0,\quad Ps^2<\infty. \]
Definition. A functional \(\Psi(P)\) is pathwise differentiable at \(P\) if there exists \(\phi_P\in L_0^2(P)\) such that for every regular submodel, \[ \left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} \Psi(P_\varepsilon)\right|_0 =P(\phi_Ps). \] Any such \(\phi_P\) in the tangent space is an influence function; the canonical gradient is the one in the closure of the tangent space with minimum variance.
Efficient influence function for a treatment-specific mean
Theorem. In the nonparametric observed-data model, \[ \Psi_a(P)=\mathbb E\{m_a(X)\} \] has efficient influence function \[ \phi_a(O)= \frac{\mathbf1(A=a)}{p_a(X)} \{Y-m_a(X)\}+m_a(X)-\Psi_a(P), \] where \(p_1=e\) and \(p_0=1-e\).
Proof. Factor \(p(o)=p(y\mid a,x)p(a\mid x)p(x)\), so every score decomposes orthogonally as \(s=s_Y+s_A+s_X\). Perturbing \(p(y\mid a,x)\) differentiates \(m_a(x)\): \[ \dot m_a(x)= \mathbb E[\{Y-m_a(x)\}s_Y\mid A=a,X=x]. \] Weighting by \(\mathbf1(A=a)/p_a(X)\) reproduces its integral against \(P_X\). Perturbing \(P_X\) contributes \(P[\{m_a(X)-\Psi_a\}s_X]\). The functional does not depend on the treatment mechanism, and the candidate is orthogonal to \(s_A\). Summing gives \(P(\phi_as)\) for every score.
Corollary. The ATE efficient influence function is \[ \phi_\tau(O)= \frac{A\{Y-m_1(X)\}}{e(X)} -\frac{(1-A)\{Y-m_0(X)\}}{1-e(X)} +m_1(X)-m_0(X)-\tau. \]
The efficiency bound
Theorem. Every regular asymptotically linear estimator of \(\tau\) in the nonparametric model has asymptotic variance at least \[ V_{\mathrm{eff}}=P\phi_\tau^2. \] An estimator with expansion \[ \widehat\tau-\tau=P_n\phi_\tau+o_P(n^{-1/2}) \] attains the bound.
Proof outline. Any regular estimator's influence function must represent the same pathwise derivative. Its projection onto the tangent space is the canonical gradient \(\phi_\tau\). Orthogonal projection gives \(\|\phi\|_2^2=\|\phi_\tau\|_2^2+ \|\phi-\phi_\tau\|_2^2\ge\|\phi_\tau\|_2^2\). The stated expansion and central limit theorem attain equality.
Variance decomposition
Proposition. \[ V_{\mathrm{eff}}= \mathbb E\left[ \frac{\sigma_1^2(X)}{e(X)} +\frac{\sigma_0^2(X)}{1-e(X)} +\{\tau(X)-\tau\}^2 \right]. \]
Proof. Conditional on \(X\), the two residual terms are mutually exclusive, centered, and have variances \(\sigma_1^2/e\) and \(\sigma_0^2/(1-e)\). The remaining conditional mean is \(\tau(X)-\tau\). Apply total variance.
One-step construction
Definition. Given an initial plug-in \(\Psi(\widehat P)\), the one-step estimator is \[ \widehat\Psi_{\mathrm{1step}} =\Psi(\widehat P)+P_n\phi_{\widehat P}. \]
For the ATE this simplifies algebraically to AIPW. The correction estimates the first-order plug-in bias. Its residual bias is the second-order remainder controlled by products of nuisance errors.
Von Mises expansion
Theorem. For a pathwise differentiable functional, \[ \Psi(\widehat P)-\Psi(P) =-P\phi_{\widehat P}+\operatorname{Rem}_2(\widehat P,P) \] under the convention \(\widehat P\phi_{\widehat P}=0\). Therefore \[ \widehat\Psi_{\mathrm{1step}}-\Psi(P) =(P_n-P)\phi_P+ (P_n-P)(\phi_{\widehat P}-\phi_P) +\operatorname{Rem}_2. \]
Proof outline. Integrate the pathwise derivative along a smooth path from \(P\) to \(\widehat P\) and isolate the derivative at one endpoint. Add the empirical influence correction, then add and subtract \((P_n-P)\phi_P\). For the ATE, direct conditional-expectation algebra yields the exact product remainder.
Nonregular targets
Example. The optimal treatment value \[ \Psi(P)=\mathbb E\max\{m_1(X),m_0(X)\} \] is not pathwise differentiable at laws assigning positive probability to \(m_1(X)=m_0(X)\). The maximizing treatment changes discontinuously under local perturbations. Ordinary influence-function normal inference can fail without a margin condition or a smoothed target.
Influence curves as diagnostics
Estimated influence values reveal which observations dominate uncertainty, especially under weak overlap. A few enormous values indicate that nominal asymptotic normality may be fragile. Their empirical variance estimates asymptotic variance, but this estimate is reliable only if nuisance remainder and tail conditions are controlled.
For the general tangent-space and convolution theory behind efficiency bounds, see the semiparametric inference interface.