Causal mediation targets with a high-dimensional mediator

CI-37 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Let $X\in[0,1]^d$, $W\in\{0,1\}$, and $M\in[0,1]^p$. Potential outcomes $Y(w,m)=F(w,m,X,U)$ have a jointly measurable representation for a specified feasible joint mediator intervention, with $U\perp(W,M)\mid X$. Assume consistency, no interference, and positivity for the intervention support. Impose bounded outcomes and a scientifically fixed feature map $h:[0,1]^p\to[0,1]^r$, $r\ll p$. Assume the controlled causal mean satisfies $\E[Y(w,m)\mid X=x]=\bar\mu_w(x,h(m))$ for every feasible $m$, with $\bar\mu_w\in\mathcal H_{d+r}^\beta(L)$. The reference distribution $g_0(dm\mid x)$ is specified or identified from a natural-mediator arm, and induced feature densities belong to $\mathcal H_{d+r}^\gamma(L)$ and are bounded below; treatment propensity belongs to $\mathcal H_d^\alpha(L)$ with fixed overlap.

Formal question

For $\vartheta\in\R^r$ in a fixed compact set, define a feasible stochastic intervention whose draw $G_\vartheta$ is independent of the individual response object conditional on $X$: \[ g_\vartheta(dm\mid x)= \frac{\exp\{\vartheta^\top h(m)\}g_0(dm\mid x)} {\int\exp\{\vartheta^\top h(u)\}g_0(du\mid x)}, \quad \psi_w(\vartheta)=\E[Y(w,G_\vartheta)-Y(w,G_0)]. \] Characterize sufficient and necessary mean-response restrictions for the target to depend only on $(X,h(M))$, rather than the full mediator. Determine sharp minimax risk and honest simultaneous inference over $\vartheta$ under the specified class. Quantify whether nuisance estimation can still incur ambient-$p$ complexity despite a response depending only on $r$ features.

Scope and status

The feature map, tilt intervention, and rate question are editorial specifications. A low-dimensional statistic of observed mediators need not be a manipulable mediator or preserve causal responses within its fibers. Identification requires a defined intervention on the original mediator, not an unexplained operation on its embedding. The target is an interventional mediated contrast; it is not automatically a natural indirect effect or a decomposition of the total treatment effect.

References for the source of the problem

Carlos Cinelli, Avi Feller, Guido Imbens, Edward Kennedy, Sara Magliacane, and Jose Zubizarreta, Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1 [stat.ME], 23 August 2025, §3.4.3, "Representing complex mediators." https://arxiv.org/html/2508.17099v1#S3.SS4.SSS3.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.