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Econ-218 · economics · Economics

Detection limits for rare long-range causal cascades

LLM status Open unverified

Statement

CI-10 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Fix a known directed network $G_N$, a source vertex $s$, and a diffusion horizon $K$. In each of $m$ independent outbreaks, source treatment $W_s\sim\operatorname{Bernoulli}(1/2)$ adds an initial infection at $s$ and is independent of all cascade randomness. Other initial infections follow a declared background-seeding distribution. Each directed edge transmits independently with unknown probability $p_e$ when first activated; the observed $Y_v\in\{0,1\}$ records infection by time $K$. This defines a finite independent-cascade structural model and counterfactuals $Y_v(1),Y_v(0)$ coupled through the same seeds and transmission randomness.

For $D_L=\{v:d_{G_N}(s,v)\geq L,\ d_{G_N}(s,v)<\infty\}$, define $\delta_v=\E[Y_v(1)-Y_v(0)]$. Restrict background probabilities and transmission parameters to declared compact sets. A rare-cascade subclass additionally bounds the probability that source treatment changes any outcome in $D_L$ by $\rho$.

Formal question

For nonempty $D_L$ and $\alpha,\beta\in(0,1)$ with $\alpha+\beta<1$, determine the minimax separation $\delta$ allowing a measurable test $\phi$ with \[ \sup_{P:\delta_v=0\ \forall v\in D_L}\Pr_P(\phi=1)\leq\alpha, \qquad \sup_{P:\max_{v\in D_L}\delta_v\geq\delta}\Pr_P(\phi=0)\leq\beta. \] Both suprema are over the declared cascade class. Determine dependence on $m,N,L,K,\rho$, graph geometry, and sparse numbers of affected vertices, with matching lower bounds and explicit nonempty alternative regimes.

Scope and status

Background infection prevents interpreting every distant infection as a treatment-induced cascade. Randomization identifies marginal treatment contrasts, but power can vanish when changed outcomes are too rare. This concrete testing problem refines the attached empirical challenge; no claim is made that its entire parameter space is an established unsolved conjecture.

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 Incorporating substantive theory. 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-218-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-218

Detection limits for rare long-range causal cascades: rare signals versus noisy long-range outcomes

Scope

The question concerns a randomized source in an independent-cascade network and distant outcome contrasts \[ \delta_v=E[Y_v\mid W=1]-E[Y_v\mid W=0],\qquad v\in D_L=\{v:\operatorname{dist}(v,\text{source})\ge L\}. \] Here outbreaks are independently repeated $m$ times. Independent-cascade monotonicity gives $\delta_v\ge0$. The rare-cascade condition bounds the probability that any distant outcome changes by $\rho$, hence $\delta_v\le\rho$. The discussion of substantive theory in Section 3.2.3 of [agenda] motivates this setting. The note shows that rarity alone does not determine the detection rate: background activation changes the experiment qualitatively.
Proposition (A general simultaneous test): Suppose $W\sim\operatorname{Bernoulli}(1/2)$ in each independent outbreak, all distant binary outcomes are observed, and $J=|D_L|\ge1$. Set $Z_v=2(2W-1)Y_v$. The test rejecting when \[ \max_{v\in D_L}\overline Z_v> t_\alpha=\sqrt{\frac{8\log(J/\alpha)}m} \] has size at most $\alpha$ against the null $\delta_v=0$ for every $v$. Its power is at least $1-\beta$ if $\max_v\delta_v\ge t_\alpha+\sqrt{8\log(1/\beta)/m}$.
Proof: $Z_v\in[-2,2]$ and $EZ_v=\delta_v$. Hoeffding's inequality gives $P(\overline Z_v-\delta_v\ge t)\le\exp(-mt^2/8)$ and the analogous lower-tail bound. A union bound proves size control, and the lower-tail bound at one coordinate attaining the stated signal proves power. Dependence among vertices within an outbreak is irrelevant to these arguments. ∎
If the allowed cascade duration $K$ is less than $L$, all such contrasts are zero. Even when $K\ge L$, the stated sufficient separation is attainable inside the rare class only when it does not exceed $\rho$.
Proposition (A noiseless path has a $1/m$ boundary): Consider a directed path of length $L$, with only its endpoint in $D_L$, all earlier transmission probabilities equal to one, and last-edge transmission probability $p$. There are no exogenous endpoint activations and $K\ge L$. Testing $p=0$ against $p>0$, rejection on the first treated endpoint activation has size zero and type-II error $(1-p/2)^m$. Every test of size at most $\alpha$ has power at most $\alpha+1-(1-p/2)^m$. Thus, for fixed $\alpha+\beta<1$, separation of order $1/m$ is both necessary and sufficient for power $1-\beta$.
Proof: In each outbreak an endpoint activation occurs with probability $p/2$. Under the null it never occurs. All earlier path outcomes are deterministic functions of $W$ and contain no information about $p$. On data with no endpoint activation, the alternative probability of each treatment sequence is the null probability multiplied by $(1-p)^{\#\{W_i=1\}}$, which is at most one. Consequently the overlap of the two experiment measures is exactly $(1-p/2)^m$, and their total variation is $1-(1-p/2)^m$. The testing inequality gives the claimed power bound. Solving the sufficient and necessary inequalities gives respectively \[ p\ge2(1-\beta^{1/m}),\qquad p\ge2\{1-(\alpha+\beta)^{1/m}\}. \] Both are of order $1/m$ for fixed error levels. Since the rare-cascade probability is $p$, this regime requires $m\rho$ bounded below. ∎
Proposition (Background noise restores a square-root boundary): In the same path, give the endpoint an independent exogenous activation with probability $b\in(0,1)$, fixed away from the endpoints. Its outcome probabilities are $b$ under control and $b+\delta$ under treatment, where $\delta=(1-b)p$. For sufficiently small $\delta$, the minimax detection separation has order $m^{-1/2}$.
Proof: The preceding simultaneous test with $J=1$ provides the upper bound. The relative entropy of one alternative outbreak against the null is $\frac12\operatorname{kl}(b+\delta,b)\le\delta^2/[2b(1-b)]$. For $m$ observations it is at most a constant times $m\delta^2$. Pinsker's inequality bounds total variation by the square root of half this quantity. If $\delta=c/\sqrt m$ with $c$ small enough, no level-$\alpha$ test has power $1-\beta$. This proves the lower bound and again requires $\delta\le\rho$. ∎
Proposition (Searching over distant leaves costs a logarithm): Let a deterministic path of length $L-1$ lead to $J\ge2$ endpoint leaves. Each leaf has an independent exogenous activation of probability $b\in(0,1)$; all last-edge probabilities are zero under the null. Under alternative $j$, only last edge $j$ is active and its endpoint contrast is $\delta$. For fixed error levels with $\alpha+\beta<1$, the separation is of order $\sqrt{\log J/m}$ whenever this quantity is sufficiently small and no larger than $\rho$.
Proof: The upper bound follows from the first proposition. Write $L_j$ for the likelihood ratio of alternative $j$ against the null over all $m$ outbreaks. Conditional on $W$, different endpoint outcomes are independent. When $W=0$ the single-outbreak likelihood ratio is one; when $W=1$, its centered second moment for the active leaf is $\delta^2/[b(1-b)]$. Therefore \[ E_0L_jL_k=1\ (j\ne k),\qquad E_0L_j^2=\left(1+\frac{\delta^2}{2b(1-b)}\right)^m. \] The uniform mixture over alternatives has chi-square divergence \[ \frac{(1+\delta^2/[2b(1-b)])^m-1}{J}. \] Take $\delta^2=c\log J/m$. For sufficiently small $c$, this divergence, and hence mixture total variation, is smaller than $1-\alpha-\beta$, uniformly for $J\ge2$. A test with power at least $1-\beta$ at every alternative would have the same power against their mixture, contradicting the testing inequality. All earlier observed vertices depend only on the treatment and do not change this calculation. ∎

Remaining gap

These are admissible path and fork submodels with exact likelihood calculations. They show why a general answer must distinguish background noise, the number of candidate locations, and the probability of a cascade. They do not characterize arbitrary graph geometry, shared transmission bottlenecks, unknown background rates, or aggregate alternatives involving many weak distant effects. The full graph-dependent minimax characterization remains unresolved.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt gives a general simultaneous test and matching detection boundaries for noiseless paths, noisy paths and searches over distant leaves. The full minimax characterization still lacks arbitrary graph geometry, shared bottlenecks, unknown background laws and alternatives with multiple weak affected vertices.

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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