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Econ-237 · economics · Economics

Network-population validity without independent unit sampling

LLM status Open unverified

Statement

CI-15 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Consider finite populations $(G_N,X_N)$ of $N$ vertices with bounded potential outcomes and a specified policy $g$. Define the population target \[ \theta_N(g)=\frac1N\sum_{i=1}^N \E[Y_i(W^g)\mid G_N,X_N]. \] A dependent sample $S_N$ has known inclusion probabilities $\pi_i>0$ and pairwise probabilities $\pi_{ij}$. Sampling is independent of potential outcomes and assignment randomization conditional on $(G_N,X_N)$. Assume randomized assignment, a valid radius-$r_N$ exposure mapping, and policy-to-design exposure density ratios bounded by $B_N$. The true neighborhoods needed for this mapping must be observed, including relevant edges to unsampled vertices.

Let $Z_i(g)$ denote an observable inverse-exposure score satisfying $\E[Z_i(g)\mid G_N,X_N]=\E[Y_i(W^g)\mid G_N,X_N]$. Declare a dependence class for these bounded scores, such as a dependency graph of maximum degree $d_N$.

Formal question

Characterize conditions on $(\pi_i,\pi_{ij},B_N,d_N)$ under which the weighted estimator \[ \widehat\theta_N(g)=\frac1N\sum_{i\in S_N}\frac{Z_i(g)}{\pi_i} \] converges uniformly to $\theta_N(g)$, and determine sharp risk and coverage rates. For a distinct target population or a limiting rooted-neighborhood distribution, characterize the additional invariance and continuity assumptions making source-network evidence transportable. Include regimes with increasing exposure radius and persistent common shocks.

Scope and status

Finite-population consistency and transport to another network are distinct questions. Local graph convergence alone need not preserve outcomes driven by global interference, and a sparse observed graph does not exclude latent common shocks. Without sampling support or invariant response mechanisms, external validity need not be identified. This is an editorial version of the attachment's dependent-sampling challenge.

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, 23 August 2025, Section 3.2.3, paragraph Statistical power, discussion of replication and external validity. https://arxiv.org/html/2508.17099v1.

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.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-237-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-237

Network-population validity without independent unit sampling: pairwise sampling structure and local transport

Scope

For a finite network population, suppose an unbiased policy score $Z_i$ is available for unit $i$ when sampled. Write $\mu_i=EZ_i$ and \[ \Psi_N=N^{-1}\sum_{i=1}^N\mu_i,\qquad \widehat\Psi_N=N^{-1}\sum_{i=1}^N\frac{I_iZ_i}{\pi_i}, \quad \pi_i=P(I_i=1)>0. \] The sampling vector is independent of the score vector, but neither vector need have independent coordinates. Section 3.2.3 of [agenda] raises replication and external validity for network populations. The following exact variance decomposition shows what first-order sampling positivity leaves uncontrolled.
Proposition (A dependent-sampling variance identity): Let $\pi_{ij}=P(I_i=I_j=1)$, with $\pi_{ii}=\pi_i$, and assume finite second moments. Then $E\widehat\Psi_N=\Psi_N$ and \[ \operatorname{Var}(\widehat\Psi_N)=\frac1{N^2}\sum_{i,j} \left[\frac{\pi_{ij}}{\pi_i\pi_j}\operatorname{Cov}(Z_i,Z_j) +\left(\frac{\pi_{ij}}{\pi_i\pi_j}-1\right)\mu_i\mu_j\right]. \] Suppose $|Z_i|\le B$, $|\mu_i|\le1$, and covariance vanishes for distinct nonadjacent vertices of a graph $H$. A uniform variance bound is \[ V_N=\frac{B^2}{N^2}\left\{\sum_i\pi_i^{-1} +\sum_{(i,j)\in E_{\rm ord}(H)}\frac{\pi_{ij}}{\pi_i\pi_j}\right\} +\frac1{N^2}\sum_{i,j}\left|\frac{\pi_{ij}}{\pi_i\pi_j}-1\right|. \] In particular $V_N\to0$ is sufficient for uniform mean-square consistency, and $\widehat\Psi_N\pm\sqrt{V_N/\alpha}$ has coverage at least $1-\alpha$.
Proof: Independence of sampling and scores gives $E[I_iZ_i/\pi_i]=\mu_i$ and $E[I_iI_jZ_iZ_j/(\pi_i\pi_j)]=\pi_{ij}E[Z_iZ_j]/(\pi_i\pi_j)$. Subtract $\mu_i\mu_j$ and sum to get the identity. Each score covariance has absolute value at most $B^2$ by Cauchy--Schwarz and is zero off the diagonal and the stated edges. Bound the remaining terms by $|\mu_i\mu_j|\le1$. Finally apply Chebyshev's inequality. The graph assumption only requires zero covariance, so a full dependency-graph assumption is stronger than needed here. ∎
Corollary (Simple random sampling): For a uniform sample of size $n$ from $N\ge2$ vertices and covariance degree at most $d$, with $B\ge1$, \[ \operatorname{Var}(\widehat\Psi_N)\le C B^2\left(\frac1n+\frac dN\right). \]
Proof: The diagonal ratio is $N/n$ and the off-diagonal ratio is $N(n-1)/\{n(N-1)\}\le1$. The diagonal covariance contribution is at most $B^2/n$, and the ordered-edge contribution is at most $B^2d/N$. The absolute sampling sum divided by $N^2$ equals $2(N-n)/(nN)\le2/n$, by separately summing diagonal and off-diagonal terms. Insert these bounds into the proposition. ∎
Example (The covariance term is real): Partition the population into equal clusters of size $h=d+1$, and let every score in cluster $c$ equal an independent mean-zero, variance-one random sign $U_c$. For simple random sampling, the estimator is the sample mean, and direct substitution in the identity gives \[ \operatorname{Var}(\widehat\Psi_N)=\frac1n+ \frac{n-1}{n}\frac d{N-1}. \] Thus even a census has variance $h/N$. This is also a minimax obstruction when the cluster signs have an unknown common mean: a census reveals only $N/h$ independent Bernoulli draws, and two means separated by $c\sqrt{h/N}$ have bounded relative entropy. The two-point argument gives squared risk at least a constant times $h/N$. With one cluster the effective information stays bounded despite $n\to\infty$.
Example (Sampling dependence can defeat a large sample): Take two deterministic equal-sized groups, with scores zero in one and one in the other. Sample exactly one entire group, each with probability $1/2$. Every unit has inclusion probability $1/2$ and $n=N/2\to\infty$, but the Horvitz--Thompson estimate is zero or one, while $\Psi_N=1/2$. Its variance is $1/4$. Here outcome dependence is absent; the second term of the exact identity records the failure.
Proposition (A quantitative local transport condition): Let $\nu_A^R,\nu_B^R$ be the laws of rooted radius-$R$ network neighborhoods in source and target populations, in a common metric space. Suppose their policy means can each be represented, up to absolute error $u_R$, by the expectation of the same $L_R$-Lipschitz function $f_R$ of that neighborhood. If the Wasserstein distance is finite, then \[ |\Psi_A-\Psi_B|\le2u_R+L_R W_1(\nu_A^R,\nu_B^R). \]
Proof: For any coupling $(H_A,H_B)$ of the two neighborhood laws, Lipschitz continuity gives $|Ef_R(H_A)-Ef_R(H_B)|\le L_REd(H_A,H_B)$. Add the two approximation errors and take the infimum over couplings. The assertion also follows by approximation if no optimal coupling exists. ∎
This bound adds a deterministic transport-bias allowance to the sampling interval. Local convergence without a uniform approximation error is insufficient: a single long cycle and two disjoint long cycles have identical fixed-radius rooted neighborhoods once the cycles are long enough, but the bounded outcome "the entire graph is connected" equals one in the former and zero in the latter. Such a global response violates the local approximation hypothesis.

Remaining gap

The variance bound need not be sharp when covariances have helpful signs, and unbiased causal scores themselves require valid exposure and assignment assumptions. The transport result requires a common response mechanism and quantitative locality; it does not derive either from network convergence. Sharp minimax inference under arbitrary graph dependence and sampling designs remains unresolved.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves an exact dependent-sampling variance identity, uniform consistency and coverage bounds, and a quantitative local-network transport condition with counterexamples. Sharp minimax inference for arbitrary inclusion designs, increasing exposure radii and general dependence still requires stronger characterization of causal scores and invariant responses.

References

  • [agenda] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane, and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2025). https://arxiv.org/html/2508.17099v1.

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