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Econ-495-A1 · revision r1 · on Econ-495

GPT 6 Astra Ultra attempt on Econ-495

Exploration · LLM-generated · gpt 6 astra ultra ·

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Contents
  1. Distributional policy effects in independent networks Sharp population-level rates and explicit conditions for node-level gains
  2. The effective sample size is a model property
  3. A no-log upper bound for weighted empirical CDFs
  4. Restrictions that make additional nodes informative
  5. Shared weights and composition cannot be ignored
  6. Quantiles and contrasts
  7. Completion Estimate
  8. References

Distributional policy effects in independent networks Sharp population-level rates and explicit conditions for node-level gains

Problem ID: C31-491. Status: Unresolved; rigorous results in explicitly declared models.

The effective sample size is a model property

The target is the mixture of node marginal outcome distributions under a supplied policy, not the distribution of individual treatment effects. We prove a $K^{-1}$ minimax squared sup-norm risk under arbitrary within-population dependence and a fixed bounded policy likelihood ratio. The result has no logarithmic factor and holds for every population size $m$. We also give explicit block and dependency-graph restrictions under which more nodes improve precision, together with simultaneous CDF and quantile conclusions.

Bhadra and Schweinberger [bs], revised Section 8.5, was inspected. Its call to study distributional features beyond means motivates the question; the independent-population benchmark and the rate proofs below are separate derivations.

Observe $K$ independent, identically distributed populations. Within population $k$, all $m$ bounded outcomes, covariates and assignment are observed. Let $R_k=q(W_k\mid G_k,A_k)/p(W_k\mid G_k,A_k)$, with $0\leq R_k\leq C$ and ${\mathbb{E}}[R_k\mid G_k,A_k]=1$. Positivity of $p$ on the support of $q$, randomized assignment independent of vector potential outcomes given $(G,A)$, and consistency imply \[ F^q(y)={\mathbb{E}}[R_kH_k(y)],\qquad H_k(y)=\frac1m\sum_{i=1}^m{\mathbf 1}\{Y_{ki}\leq y\}. \tag{2} \] Indeed, condition on $(G,A)$ and all potential outcomes, sum over assignment vectors, cancel $p$ in $p(q/p)$, and then integrate. This uses the full assignment vector and permits unrestricted interference within populations.

A no-log upper bound for weighted empirical CDFs

Suppose the known outcome range is $[a,b]$. Define the raw estimator $\overline F(y)=K^{-1}\sum_kR_kH_k(y)$. It is monotone but its total mass need not be exactly one. Form a proper CDF $\widehat F$ by clipping $\overline F$ to $[0,1]$ for $a\leq y<b$, putting zero for $y<a$ and one for $y\geq b$. This adds any missing terminal mass at the known upper endpoint. It does not increase sup-norm error relative to $F^q$.
Lemma (Weighted threshold process): Let $(R_j,V_j)$, $j=1,\ldots,N$, be independent, not necessarily identically distributed, with $0\leq R_j\leq C$ and ${\mathbb{E}} R_j=1$. Then \[ {\mathbb{E}}\sup_y\left|\frac1N\sum_j \{R_j{\mathbf 1}\{V_j\leq y\}-{\mathbb{E}}[R_j{\mathbf 1}\{V_j\leq y\}]\}\right|^2 \leq\frac{16C}{N}. \]
Proof: Use independent copies $Z'_j(y)$ of $Z_j(y)=R_j{\mathbf 1}\{V_j\leq y\}$. Jensen's inequality for the convex functional $\|\cdot\|_\infty^2$ bounds the centered process's expected squared norm by that of $N^{-1}\sum_j(Z_j-Z'_j)$. Symmetry of each difference lets us insert independent Rademacher signs $\xi_j$. The squared triangle inequality then bounds it by $4N^{-2}{\mathbb{E}}\|\sum_j\xi_j Z_j\|_\infty^2$. Conditional on $(R_j,V_j)$, order the observations by $V_j$, breaking ties deterministically. The threshold sums are among the partial sums of the independent mean-zero increments $\xi_jR_j$. Doob's $L^2$ maximal inequality bounds the conditional expected squared maximum by $4\sum_jR_j^2$. One may verify that inequality directly: for a square-integrable martingale $S_l$, stopping at its first crossing of $t$ gives $t\Pr(\max|S_l|>t)\leq{\mathbb{E}}[|S_N|{\mathbf 1}\{\max|S_l|>t\}]$; integration and Cauchy--Schwarz give ${\mathbb{E}}\max|S_l|^2\leq4{\mathbb{E}}|S_N|^2$. Finally $R_j^2\leq CR_j$, so $\sum_j{\mathbb{E}} R_j^2\leq NC$. Combining the factors proves the result. All suprema are measurable because the sample threshold paths are right-continuous and the same supremum can be taken over a countable dense set including the endpoints. ∎
Theorem (Risk and a simultaneous band): For every $K,m$ in the declared experiment, \[ {\mathbb{E}}\|\widehat F-F^q\|_\infty^2\leq\min\{1,16C/K\}. \] For any $\alpha\in(0,1)$ the CDF band \[ [\max(0,\widehat F(y)-r_K),\min(1,\widehat F(y)+r_K)] \] covers $F^q(y)$ simultaneously for all $y$ with probability at least $1-\alpha$, where \[ r_K=\min\left\{1,4\sqrt{C/K} +C\sqrt{\frac{\log(1/\alpha)}{2K}}\right\}. \]
Proof: Independently select a uniform node index $J_k$ in each observed population. The pairs $(R_k,Y_{kJ_k})$ are independent across $k$ and have ${\mathbb{E}} R_k=1$. Their weighted empirical CDF has mean $F^q$ by (2). Its conditional expectation given all the observed populations is $\overline F$. Conditional Jensen and Lemma Lemma (Weighted threshold process) give expected squared sup error at most $16C/K$ for $\overline F$, and clipping preserves the bound for $\widehat F$. The bound one follows because both $\widehat F$ and $F^q$ are CDFs.

For the raw error $Z=\|\overline F-F^q\|_\infty$, Cauchy--Schwarz gives ${\mathbb{E}} Z\leq4\sqrt{C/K}$. Replacing one entire population changes $R_kH_k(y)$ by at most $C$ at every $y$, so it changes $Z$ by at most $C/K$. The bounded-differences inequality yields $\Pr\{Z>{\mathbb{E}} Z+t\}\leq\exp(-2Kt^2/C^2)$. Use the displayed square-root value of $t$ and the pointwise clipping argument. If the resulting radius exceeds one, the radius-one band covers surely. ∎

Randomly selecting one node is only a proof device: the estimator uses every observed node and is the conditional average of that less efficient procedure. No within-population independence or graph sparsity has entered the proof.

Theorem (Matching lower bound): If the model allows arbitrary common within-population outcome shocks, its minimax squared sup-norm risk is bounded below by $c/K$ for a numerical $c>0$, independently of $m$. For fixed $C$, Theorem Theorem (Risk and a simultaneous band) is therefore rate optimal.
Proof: Take outcomes in $\{0,1\}$ and set all potential outcomes of all nodes in a population equal to a single Bernoulli variable $B_k$ independent of assignment and covariates. Let its success probability be $\theta$. Every target policy has $F^q(1/2)=1-\theta$, and the entire population contributes just one Bernoulli observation about $\theta$. Compare $\theta_-=1/2-h$ and $\theta_+=1/2+h$ with $h=c_0/\sqrt K$. Their $K$-sample KL divergence is at most $C_0Kh^2$, so choosing $c_0$ small gives total variation at most $1/2$. For any CDF estimator, decode the nearer of the two CDF values at $1/2$. On a testing error its squared sup loss is at least $h^2$. The sum of the two testing errors is at least $1/2$, so the maximum risk is at least $h^2/4$. Assignment data have the same independent law in both models and cannot improve this test. ∎

Restrictions that make additional nodes informative

A useful sufficient restriction must concern the full observed weighted contributions. Conditional independence of outcomes given a shared graph, assignment or environment is not enough by itself.
Proposition (Independent interference blocks): Suppose each population is partitioned into $B=m/b_0$ equal-sized blocks, there is no interference across blocks, and the observed blocks, including all their covariates and assignments, are independent across all $KB$ blocks. Suppose the observed and target policies factor over blocks and their block likelihood ratios satisfy $0\leq R_{kr}^{\rm block}\leq C_0$, with conditional mean one. Assume these policies also satisfy the original population likelihood-ratio restriction when remaining in the canonical model. Then the estimator averaging the block-weighted empirical CDFs has risk at most \[ 16C_0/(KB)=16C_0b_0/(Km). \] If the class permits identical outcomes within each block, its minimax rate for fixed $C_0$ is exactly $b_0/(Km)$.
Proof: No cross-block interference and policy factorization give the analogue of (2) for each block using its local ratio; weights for other blocks integrate to one. Equal block sizes mean that averaging these targets equals the target average across all nodes. Apply the independent-node-selection proof of Theorem Theorem (Risk and a simultaneous band) to the $KB$ independent blocks, using Lemma Lemma (Weighted threshold process), which permits different block distributions. The lower bound uses an independent Bernoulli outcome common to the nodes of each block and invariant under policy, reducing to $KB$ Bernoulli observations as in Theorem Theorem (Matching lower bound). The submodel $q=p$ meets both the global and block ratio conditions. ∎

A more flexible restriction can be expressed directly through local policy contributions. Let $N=Km$ and suppose there are valid local weights $0\leq R_i\leq C_0$, ${\mathbb{E}} R_i=1$, such that the target is $N^{-1}\sum_i{\mathbb{E}}[R_i{\mathbf 1}\{Y_i\leq y\}]$. Assume the pairs $(R_i,Y_i)$ have a supplied dependency graph with a proper coloring into $\chi$ independent color classes. Independence means joint independence of every collection inside a color class, including covariates and weights; sparse observed friendship edges alone do not imply it. For a class of size $N_l$, Lemma Lemma (Weighted threshold process) bounds its centered-process root mean squared sup norm by $4\sqrt{C_0/N_l}$. The triangle inequality in $L^2$ and Cauchy--Schwarz give \[ \left({\mathbb{E}}\|\overline F-F^q\|_\infty^2\right)^{1/2} \leq\frac{4\sqrt{C_0}}N\sum_{l=1}^\chi\sqrt{N_l} \leq4\sqrt{\frac{C_0\chi}{N}}. \] Clipping gives a proper CDF as before. Markov's inequality supplies a simultaneous band of radius $\min(1,4\sqrt{C_0\chi/(N\alpha)})$. Thus $\chi=o(Km)$ suffices for shrinking bands under these explicit restrictions. We do not assert a matching minimax rate for every dependency graph.

Shared weights and composition cannot be ignored

For any shared information $V$ making the population ratio $R$ measurable, the exact variance decomposition for a fixed $y$ is \[ {\operatorname{Var}}(RH(y))={\mathbb{E}}[R^2{\operatorname{Var}}(H(y)\mid V)] +{\operatorname{Var}}(R{\mathbb{E}}[H(y)\mid V]). \] Even if node outcomes are conditionally independent, only the first term is necessarily of order $1/m$. The second can remain positive. For example, let an observed population-level assignment $W$ be a fair coin, take $q=p$, and set every node outcome deterministically equal to $W$. Outcomes are conditionally independent given $W$ (each is degenerate), but the raw empirical CDF at $1/2$ has variance $1/(4K)$. In this particular known-response example a model-aware estimator can do better, so this calculation is an obstruction to an unjustified variance argument, not a universal minimax lower bound. Theorem Theorem (Matching lower bound)'s unobserved common shock supplies the actual minimax obstruction.

If $R$ is independent of $m$ independent node outcomes with CDF $F$, the same identity gives ${\operatorname{Var}}(RH(y))={\mathbb{E}}[R^2]F(y)(1-F(y))/m+{\operatorname{Var}}(R)F(y)^2$. Self-normalization or design stratification can sometimes remove that second component. Therefore a shared weighting penalty of the raw estimator must not automatically be described as information-theoretically unavoidable.

Quantiles and contrasts

Let $Q^q(u)=\inf\{y:F^q(y)\geq u\}$ for fixed $u\in(0,1)$. Assume uniformly over the class that $F^q$ has a density at least $c_*>0$ throughout $[Q^q(u)-h_0,Q^q(u)+h_0]$, for supplied common $h_0>0$, and this interval lies inside the outcome range. Then on any event $\|\widehat F-F^q\|_\infty\leq r<c_*h_0$, \[ |\widehat Q(u)-Q^q(u)|\leq r/c_*. \] To prove it, continuity gives $F^q(Q^q(u))=u$, and integrating the density lower bound gives $F^q(Q^q(u)+t)\geq u+c_*t$ and $F^q(Q^q(u)-t)\leq u-c_*t$ for $0\leq t\leq h_0$. The sup-norm bound brackets the first point where $\widehat F$ reaches $u$, with endpoint equality obtained by a limit. A directly usable inversion interval is $[\widehat Q(u-r),\widehat Q(u+r)]$ when $r<\min(u,1-u)$, by the defining CDF inequalities.

For two policies, construct both CDF bands with failure budgets summing to $\alpha$. Their difference of quantile estimators has absolute error at most $r_1/c_1+r_0/c_0$ on the joint event. This is a contrast of marginal quantiles. No cross-world coupling is needed, and no distribution of individual policy effects is identified. The common neighborhood requirement matters: a merely positive density at one point, without a uniform neighborhood, does not imply a uniform quantile risk or interval-width guarantee.

The unrestricted independent-population benchmark is resolved at the level of rates. General dependence-decay models, optimal policy reweighting when global ratios grow, and uniform quantile results without the stated density neighborhoods remain outside these results and keep the broader target unresolved.

Completion Estimate

85%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves matching independent-population minimax rates, shrinking simultaneous distribution bands, node-level improvements under block and dependency-graph restrictions, and quantile contrasts. General dependence-decay minimax characterizations and optimal weighting beyond these specified independence structures remain unresolved, so the broader requested characterization is not fully closed.

References

  • [bs] S. Bhadra and M. Schweinberger (2026), Causal Inference Under Network Interference, arXiv:2508.06808v2, 9 August 2026, Section 8.5, "Random Potential Outcomes: More than Means," inspected 9 October 2026. https://arxiv.org/html/2508.06808v2#S8.SS5.