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Econ-127 · economics · Economics

Observational interference with observed and latent network confounding

LLM status Open unverified

Statement

CI-12 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Observe $n$ independent clusters of $N$ units, recording pretreatment network $G$, all observed covariates $X$, assignment vector $W\in\{0,1\}^N$, and outcomes $Y_i=Y_i(W)\in[0,1]$. Let $U$ contain the full latent potential-outcome table. Require both $q(w\mid G,X)>0$ and $q(w\mid G,X,U)>0$ almost surely. Impose the sensitivity restriction, for every pair $w,w'$, \[ \Gamma^{-1}\leq \frac{q(w\mid G,X,U)/q(w'\mid G,X,U)} {q(w\mid G,X)/q(w'\mid G,X)} \leq\Gamma \] almost surely, where $\Gamma\geq1$ is specified. Additional graph-local response and observed adjustment restrictions must be explicitly declared. For two specified policies $g,g'$ with bounded density ratios relative to $q$, target $\theta(P)=N^{-1}\sum_i\E_P[Y_i(W^g)-Y_i(W^{g'})]$, using policy randomization independent of the potential table.

Formal question

Characterize when the observed law and the declared network restrictions identify $\theta$, and otherwise derive the sharp identified set over the sensitivity class. Determine which own-unit and neighbor covariates form sufficient adjustment sets when $\Gamma=1$, and which exposure reductions preserve the policy target. For bounded-degree response classes, obtain optimal estimation rates and honest confidence sets for the identified set, allowing $N$ to grow with $n$.

Scope and status

At $\Gamma=1$, the restriction entails full-vector conditional exchangeability; it is an assumption, not a conclusion obtained merely by adjusting for own covariates. For $\Gamma>1$, arbitrary point identification is not demanded. Full-vector positivity may deteriorate exponentially with cluster size and must enter any rate claim. This sensitivity-based subclass retains the attachment's concern about network confounding without silently declaring all confounders observed.

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 Interference in observational studies. 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-127-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-127

A sharp finite sensitivity program and local exposure estimation

Catalogue problem: C31-124. Status: Unresolved. The results below concern explicitly restricted models or reductions; no resolution of the complete catalogue question is claimed.

Abstract

For binary cluster outcomes and finite assignment spaces, the full-vector odds-ratio sensitivity model has an exact finite linear-program description in latent potential-table probabilities. Its projected target interval is sharp, including declared graph-local restrictions. At sensitivity one, local response functions permit policy evaluation with local exposure probabilities; an explicit finite-sample bound records dependence on degree rather than on the full assignment space. A compatible binary example demonstrates nonidentification as soon as the sensitivity parameter exceeds one.

Question and finite experiment

Clusters are independent. Within one cluster, $W\in\mathcal W=\{0,1\}^N$ is the assignment vector and $Y_i=Y_i(W)$. Initially fix one observed covariate/network stratum, and assume all outcomes are binary. Write \[ p_{w,y}={\mathbb{P}}(W=w,Y=y),\qquad q_w=\sum_y p_{w,y}>0, \quad y\in\{0,1\}^N. \] Let $U=(Y_i(w):i\in[N],w\in\mathcal W)$ be the full potential table. The sensitivity restriction is, for every $w,w'$ and every potential table with positive probability, \[ \Gamma^{-1}\le \frac{{\mathbb{P}}(W=w\mid U)/{\mathbb{P}}(W=w'\mid U)}{q_w/q_{w'}} \le\Gamma, \qquad 1\le\Gamma<\infty. \tag{1} \] Policies $g,g'$ are specified probability distributions on $\mathcal W$, randomized independently of $U$. The target is \[ \theta={\mathbb{E}}\left[\sum_{w\in\mathcal W}(g_w-g'_w) \frac1N\sum_iY_i(w)\right]. \tag{2} \] No claim that observed own-unit covariates suffice for (1) with $\Gamma=1$ is made. The source explicitly identifies observational interference and network confounding as an agenda [CF].

A sharp linear program

Let $\mathcal U$ be the set of allowed binary potential tables. With unrestricted interference it has $2^{N2^N}$ elements. Declared graph-local restrictions may reduce it: for example, require $u_i(w)=u_i(w')$ whenever $w,w'$ agree on a specified neighborhood $B_i$. Such restrictions concern potential responses, not just the observed graph.

For each allowed $u$ and assignment $w$, introduce a variable $z_{u,w}$. Consider the finite system \begin{align} z_{u,w}&\ge0, \label{wsref-0-65-71-3a-6e-6f-6e-6e-65-67-61-74-69-76-65}\\ \sum_{u:u(w)=y}z_{u,w}&=p_{w,y}, &&(w,y),\label{wsref-0-65-71-3a-6f-62-73-65-72-76-65-64}\\ q_{w'}z_{u,w}&\le\Gamma q_wz_{u,w'}, &&(u,w,w').\label{wsref-0-65-71-3a-6f-64-64-73} \end{align} Here $u(w)$ is the entire outcome vector under assignment $w$, and all $q_w$ are fixed from the observed law. Define \[ c_u=\sum_w(g_w-g'_w)\frac1N\sum_iu_i(w),\qquad \Theta(z)=\sum_uc_u\sum_wz_{u,w}. \]

Theorem (Sharp finite identified interval): If the system (eq:nonnegative)--(eq:odds) is feasible, the sharp identified set for (2) under consistency, the allowed-table restrictions, strict conditional assignment positivity, and (1) is \[ [\theta_-,\theta_+],\qquad \theta_- =\min_z\Theta(z),\quad \theta_+ =\max_z\Theta(z), \tag{3} \] where both extrema are attained over the displayed system. If it is infeasible, the observed law is incompatible with the declared restrictions. Every value between the endpoints is attained by a compatible causal model.
Proof: For a compatible model, let $z_{u,w}={\mathbb{P}}(U=u,W=w)$. Consistency gives (eq:observed). On a row of positive mass $t_u=\sum_wz_{u,w}$, conditional probabilities are $z_{u,w}/t_u$; substituting them into (1) and clearing positive denominators gives exactly (eq:odds), including its reverse inequality since all ordered pairs are imposed. The target is $\Theta(z)$.

Conversely, take any feasible $z$. Summing (eq:observed) shows $\sum_{u,w}z_{u,w}=1$, so it defines a joint law of $(U,W)$. If $t_u>0$, at least one coordinate in that row is positive. Inequality (eq:odds) and $q_w>0$ force every coordinate in that row to be positive: a zero coordinate would force all the others to zero. Thus strict conditional positivity holds automatically for every realized table. Clearing denominators in reverse gives (1). Define potential outcomes from $u$, and observe $Y=u(W)$; this reproduces the observed law and all table restrictions. Generate each policy assignment independently of $U$ with probabilities $g$ or $g'$, obtaining target $\Theta(z)$.

The feasible set is a closed bounded polytope: nonnegativity and total mass one bound all coordinates. Its linear image is compact and convex in ${\mathbb{R}}$, hence an attained interval. Convex combinations of feasible $z$ obey the same constraints and achieve every intermediate objective value. This proves sharpness and attainment. ∎

This is an exact characterization for the finite binary model, not a polynomial algorithm in $N$. The number of tables is already enormous. Under neighborhood response restrictions with $|B_i|\le d+1$, the number becomes at most $2^{\sum_i2^{|B_i|}}$, still exponential in $N$ for fixed degree. Finite observed strata can be handled by one program per stratum and averaging with their observed probabilities; restrictions coupling strata need to be imposed jointly.

Sensitivity one and local exposure reduction

Proposition (Identification at $\Gamma=1$): For $\Gamma=1$, every feasible row satisfies $z_{u,w}=t_uq_w$, so $U$ and $W$ are independent in the fixed stratum. Consequently \[ \theta=\sum_w(g_w-g'_w){\mathbb{E}}[\overline Y\mid W=w], \qquad \overline Y=N^{-1}\sum_iY_i. \tag{4} \] If $Y_i(w)=y_i(E_i(w))$ for specified finite exposures $E_i$, let $q_i(e)$, $g_i(e)$, and $g_i'(e)$ be their induced exposure probabilities. Then \[ \theta=\frac1N\sum_i\sum_e \{g_i(e)-g_i'(e)\}{\mathbb{E}}[Y_i\mid E_i(W)=e]. \tag{5} \] The second formula requires the declared exposure response restriction; equality of exposure labels alone is not evidence for it.
Proof: Both directions of (eq:odds) at $\Gamma=1$ imply $z_{u,w}/q_w$ is constant in $w$. Summing over $w$ identifies that constant as $t_u$. This gives independence and, by consistency, identifies the potential mean at each full assignment. Formula (4) follows. Under the exposure restriction, independence of $U$ and $W$ implies ${\mathbb{E}}[y_i(e)\mid E_i(W)=e]={\mathbb{E}} y_i(e)$. Grouping each policy's assignment sum by its exposure gives (5). ∎
Theorem (Known-exposure-probability risk and interval): In the setting of (5), suppose the $q_i(e)$ are known, positive on all policy-supported exposures, and \[ 0\le g_i(e)/q_i(e)\le K,\qquad 0\le g_i'(e)/q_i(e)\le K. \] For independent clusters $c=1,\ldots,n$, set \[ Z_c=\frac1N\sum_i \frac{g_i(E_i(W_c))-g_i'(E_i(W_c))}{q_i(E_i(W_c))}Y_{ic}, \qquad \hat\theta=n^{-1}\sum_c Z_c. \] Then ${\mathbb{E}}\hat\theta=\theta$, ${\mathbb{E}}(\hat\theta-\theta)^2\le K^2/n$, and \[ {\mathbb{P}}\left\{|\hat\theta-\theta|\le K\sqrt{2\log(2/\alpha)/n}\right\}\ge1-\alpha. \] No independence of outcomes within a cluster is required. If assignments are independent Bernoulli with success probabilities in $[\epsilon,1-\epsilon]$ and exposures record at most $d+1$ treatment coordinates, one may take $K=\epsilon^{-(d+1)}$ for arbitrary policies.
Proof: Taking expectations of each weighted summand and using (5) proves unbiasedness. Each difference of density ratios is in $[-K,K]$ and $Y_{ic}\in[0,1]$, so $|Z_c|\le K$. Cluster independence gives variance at most $K^2/n$. For completeness, a bounded variable in an interval of length $2K$ satisfies ${\mathbb{E}} e^{\lambda(Z-{\mathbb{E}} Z)}\le e^{\lambda^2K^2/2}$; this follows from convexity of the exponential followed by maximization over the interval's mean. Multiplying this bound over independent clusters and optimizing the exponential Markov inequality gives the stated two-sided radius. Finally every local assignment pattern has probability at least $\epsilon^{d+1}$, while any policy exposure probability is at most one. ∎

Full-vector probabilities may be exponentially small in $N$, while local exposure probabilities depend on $d$. This reduction relies on the substantive local-response and exchangeability assumptions. It is not obtained by declaring neighbor treatments ignorable after own-covariate adjustment.

Nor should one expect a universal extra factor $1/N$ in the variance without assumptions on outcome dependence. For example, under independent fair treatment assignments let $Y_i(w)=w_iV$, where a common cluster variable $V$ is Bernoulli with mean $\vartheta$. The all-treated versus all-untreated contrast equals $\vartheta$, and local exposure ratios are at most two. Even observing all of $(V,W)$ in every cluster supplies only $n$ Bernoulli draws about $\vartheta$, so the two-point Bernoulli testing bound gives risk at least $c/n$. The actual data are a function of those augmented data and cannot have a smaller minimax lower bound. These deterministic policies have full-vector density ratios $2^N$; the example concerns the local-ratio theorem, not a subclass with a fixed uniform full-vector ratio bound as $N$ grows.

An explicit nonidentification example

Take $N=1$, no covariates, and observed law ${\mathbb{P}}(W=w,Y=y)=1/4$ for each binary $w,y$. Fix $\Gamma>1$. In a first model let $Y(0)$ be a fair Bernoulli variable independent of $Y(1)$, and put \[ t_+={\mathbb{P}}(Y(1)=1)=\frac{3\Gamma-1}{4\Gamma}, \quad {\mathbb{P}}(W=1\mid Y(1)=1)=\frac\Gamma{3\Gamma-1}, \quad {\mathbb{P}}(W=1\mid Y(1)=0)=\frac\Gamma{1+\Gamma}. \] Let assignment depend on no further variable. Then both joint probabilities ${\mathbb{P}}(W=1,Y(1)=1)$ and ${\mathbb{P}}(W=1,Y(1)=0)$ equal $1/4$. Independence of $Y(0)$ gives the two control cells as well. Both displayed propensities lie in $[1/(1+\Gamma),\Gamma/(1+\Gamma)]$, which is exactly (1) when $q_0=q_1=1/2$.

A second model instead takes $t_-=(\Gamma+1)/(4\Gamma)$ and interchanges the two displayed propensities for the two values of $Y(1)$. It has the same four observed cells and obeys the same sensitivity restriction. Their all-treated versus untreated effects are respectively \[ t_+-\tfrac12=\frac{\Gamma-1}{4\Gamma}, \qquad t_--\tfrac12=-\frac{\Gamma-1}{4\Gamma}. \] Thus point identification fails for every $\Gamma>1$ even before introducing network complexity. These two values are feasible witnesses; they are not asserted to be the sharp endpoints. Those endpoints are determined by (3).

Remaining estimation and graph questions

The finite program solves sharp identification in a binary, finite-stratum model. Continuous outcomes and continuous covariates need additional approximation or infinite-dimensional arguments. Estimating program endpoints uniformly when some assignment cells are rare, deriving optimal rates with $N$ increasing, learning unknown exposure propensities, and exploiting graph structure to avoid latent-table enumeration remain unresolved. In particular, the confidence interval above is a point-identification result at $\Gamma=1$ with known exposure probabilities; it is not an honest interval for sensitivity endpoints at $\Gamma>1$.

References

  • [CF] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane, and J. Zubizarreta (2025). Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference. arXiv:2508.17099v1, Section 3.2.3, "Interference in observational studies." https://arxiv.org/html/2508.17099v1.

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