Machine-learning debiased efficient estimation under generalized data-fusion alignments

Let $X$ be a latent/full-data random element with law $P$ on a measurable space $(\mathcal X,\mathcal A)$, where $P$ belongs to a semiparametric model $\mathcal P$. The estimand is a finite-dimensional parameter $\psi(P)\in\mathbb R^d$, with $\psi:\mathcal P\to\mathbb R^d$ pathwise differentiable at the true law $P_0$.

This setup follows Graham et al. (2024).

Data are collected from $K\ge 2$ independent sources. For source $k$, we observe i.i.d. data $O^{(k)}_1,\dots,O^{(k)}_{n_k}$ in $(\mathcal O_k,\mathcal F_k)$ from an observed-data law $Q_k$, with $n=\sum_{k=1}^K n_k$ and $n_k/n\to\pi_k\in(0,1)$. Each $Q_k$ is induced by $(P,\eta_k)$, where $\eta_k$ denotes source-specific nuisance features (e.g., observation/coarsening mechanisms and other non-target components). Assume the sources satisfy a prespecified set of alignment restrictions consisting of: (i) conditional alignment constraints equating selected conditional distributions under $Q_k$ to corresponding conditionals of $P$, and (ii) marginal alignment constraints equating selected marginals under $Q_k$ to corresponding marginals of $P$. These alignments may come from different factorizations of $P$ (not necessarily one common factorization), and together they identify $\psi(P)$.

Let $\mathcal Q$ be the observed-data model induced by all $(P,\eta_1,\dots,\eta_K)$ satisfying the alignments. For $\psi$, let $\varphi_{\mathrm{eff},k}\in L_2^0(Q_k)$ denote the source-indexed efficient influence function contribution for source $k$ at the truth, with $\mathbb E[\varphi_{\mathrm{eff},k}(O^{(k)})]=0$, and define

(Equivalent pooled-data notation uses $W=(S,O)$ with source indicator $S\in\{1,\dots,K\}$ and EIF $\phi_{\mathrm{eff}}(W)$.)