Econ-43 · economics · Economics

Sequential adaptive experiments with uncertain causal structure

LLM status Open unverified

Statement

CI-01 | Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Fix a finite horizon $K$, finite action spaces $\mathcal W_k$, and an observed longitudinal record $O=(X,W_1,L_1,\ldots,W_K,Y)$ with $0\leq Y\leq1$. Let $\mathcal G$ be a specified finite collection of causal directed acyclic graphs, possibly containing latent variables, and let $\mathcal M(\mathcal G)$ be the structural models obeying their restrictions. Each model must give unique measurable counterfactuals $Y^g$ for a specified intervention regime $g=(g_1,\ldots,g_K)$. The constant regime $g_k\equiv w_k$ gives the target $\theta_g(P)=\E_P[Y^g]$.

Across $n$ episodes with independent structural errors before assignment, the design may choose assignment kernels $q_{i,k}$ using previous episodes and the current observed history. Observed episodes can therefore be dependent. Assignments use external randomness independent of structural errors conditional on that history. Specify which treatment coordinates can be manipulated and their allowable probabilities; latent variables remain unrestricted except by the declared structural class.

Formal question

For each such intervention family, characterize when equality of the laws of all observable adaptive experiments implies equality of $\theta_g$. When identification holds, determine the minimax mean squared risk over $\mathcal M(\mathcal G)$, optimizing over allowable adaptive designs and estimators. When it fails, determine the sharp identified set \[ \Theta_g(Q)=\{\theta_g(P):P\in\mathcal M(\mathcal G),\ (P_{\rm obs}^{q})_{q\in\mathcal Q}=Q\} \] where $\mathcal Q$ is the allowable design family and $Q$ its family of observable laws. For any implemented design, seek informative confidence sets for its corresponding identified set. Establish matching lower bounds in terms of intervention support, graph uncertainty, and horizon.

Scope and status

Complete randomization of every needed action with adequate support already resolves many identification questions. The research target is the specified restricted-intervention class and its sharp statistical limits; ignorance of confounders cannot by itself justify point identification. This is a source-dated research agenda refinement.

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 3.1.3, paragraph Sequential treatments and increased adaptivity. 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 2

Econ-43-A2 · Exploration · LLM-generated · gpt 6 astra pro · 2026-10-09

GPT 6 Astra Pro attempt on Econ-43

{Adaptive causal experiments under graph uncertainty}

C90-42.tex

Research attempt and adversarial audit --- 9 October 2026

Tools. Web browsing: YES. Computation: YES (Python, exact integer/rational checks, and explicitly identified numerical diagnostics). Status convention. A full proof of a restricted theorem does not resolve the full input. Literature results are marked as imported; no novelty or exhaustive current-literature certification is claimed. Computational searches certify only their stated finite domains.

FORMAL RESTATEMENT

Let $K\in{\mathbb{N}}$ be a fixed finite treatment horizon, with finite action sets $\mathcal W_k$, and standard Borel observed histories. A fixed structural model $m$ belongs to a declared class $\mathfrak M(\mathcal G)$ associated with a finite DAG family $\mathcal G$; latent variables are allowed. Each model defines unique measurable counterfactuals under a target regime $g$, and $0\le Y^g\le1$. Write \[ \theta(m)={\mathbb{E}}_m[Y^g]. \] This notation corrects a potential ambiguity: before identification, $\theta$ belongs to a structural model, not to an observed-data law alone. Each of $n$ episodes draws a fresh independent copy of that model's structural errors. Assignment kernels may use previous episodes and current observed history but only external randomization conditionally independent of the structural errors. The allowable family $\mathcal Q$ specifies which such kernels can be implemented.

Define experimental equivalence by \[ m\sim_{\mathcal Q}m'\ \Longleftrightarrow\ P_m^q=P_{m'}^q\quad\text{for every allowable experiment }q. \] Identification means $m\sim_{\mathcal Q}m'\Rightarrow\theta(m)=\theta(m')$ for every $m,m'$. An observed family $Q$ has identified set $\{\theta(m):(P_m^q)_{q\in\mathcal Q}=Q\}$. For a specified design class the statistical target is \[ R_n=\inf_{q,\widehat\theta}\sup_{m\in\mathfrak M(\mathcal G)} {\mathbb{E}}_m^q[(\widehat\theta-\theta(m))^2]. \] The request asks for these objects across specified intervention families and for confidence sets and support/horizon lower bounds. Without fixing those families and structural restrictions, there is no one numerical minimax rate to prove. We supply exact results for two fully specified, contrasting families, plus a supported sequential-design bound.

Intervals, expectations, and confidence coverage refer to the actual adaptive data law. Coverage of each possible parameter and simultaneous coverage of the entire identified set are distinguished. Empty data ($n=0$) are allowed only in explicitly identified boundary checks.

QUICK LITERATURE/CONTEXT CHECK

Cinelli et al. [causal], Section 3.1.3, explicitly discuss sequential treatments and increased adaptivity. The input is a formal research agenda built around that discussion, not a theorem claiming that arbitrary graph uncertainty is automatically eliminated by adaptive experiments. The exact minimax calculations below are independently derived for their stated structural submodels; they do not purport to settle all restricted-intervention identification problems.

ATTACK PLAN

This is a causal identification and statistical decision problem. The proof track first implements the target intervention, then derives an exact bounded-mean minimax rule; it also studies inverse-probability weighting under restricted support. The disproof track creates observationally identical models with different unobserved counterfactuals. A third construction varies action support while leaving the graph family fixed.

Tool	Why relevant
Experimental equivalence	States exactly what identification requires.
Direct intervention	Removes graph ambiguity when the target is implementable.
Beta--Bernoulli decision theory	Gives an exact, not merely order-wise, minimax lower bound.
Conditional change of measure	Justifies sequential weighting despite latent variables.
Martingale orthogonality	Handles dependence introduced across episodes.
Two indistinguishable endpoints	Proves nonidentification and positive minimax risk.
Tonelli's theorem	Converts common-law confidence coverage into a length lower bound.
Variance inequalities	Supply finite-sample confidence guarantees without asymptotics.

WORK

Fully implementable target: an exact minimax theorem

Theorem: Assume every episode may be assigned regime $g$ itself. Assume the model class contains the following submodel for every $\theta\in[0,1]$: all observable variables other than $Y$ are fixed or generated by the design, and $Y\sim\operatorname{Bernoulli}(\theta)$ independently of every action and independently across episodes. Then, for $n\ge1$, \[ R_n=\frac{1}{4(\sqrt n+1)^2}. \] An optimal design always implements $g$, and an optimal estimator is \[ \widehat\theta_n=\frac{\sum_{i=1}^nY_i+\sqrt n/2}{n+\sqrt n}. \] In this design $\theta$ is identified by ${\mathbb{E}}[Y]$.

Proof: Under regime $g$, consistency and independent replicated structural errors make the outcomes independent copies of $Y^g$. Put $\theta={\mathbb{E}}[Y^g]$. Since $0\le Y^g\le1$, $(Y^g)^2\le Y^g$, so ${\operatorname{Var}}(Y^g)\le\theta(1-\theta)$. The estimator has variance $n{\operatorname{Var}}(Y^g)/(n+\sqrt n)^2$ and bias $\sqrt n(1/2-\theta)/(n+\sqrt n)$. Its mean squared error is therefore at most \[ \frac{n\{\theta(1-\theta)+(1/2-\theta)^2\}}{(n+\sqrt n)^2} =\frac1{4(\sqrt n+1)^2}. \]

For the lower bound, restrict to the stated Bernoulli submodel and put a proper $\operatorname{Beta}(a,a)$ prior on $\theta$, where $a=\sqrt n/2>0$. For any adaptive design, the joint likelihood factors into $\theta^{S}(1-\theta)^{n-S}$, $S=\sum Y_i$, times factors for assignments and auxiliary randomization that are independent of $\theta$ given the observed past. Thus the posterior density is proportional to $\theta^{S+a-1}(1-\theta)^{n-S+a-1}$. Its mean is $(S+a)/(n+2a)$; the identity follows by integrating the beta density, or by integrating the derivative of $\theta^{S+a}(1-\theta)^{n-S+a}$ over $[0,1]$.

For squared error, conditional posterior risk of an estimate $d$ is ${\operatorname{Var}}(\theta\mid\text{data})+(d-{\mathbb{E}}[\theta\mid\text{data}])^2$, so this posterior mean is Bayes optimal. In the Bernoulli submodel the variance bound in the upper calculation is an equality for every $\theta$; hence this rule has constant frequentist risk equal to the displayed value. Its integrated Bayes risk is that value. The supremum risk of every other rule and every design is at least its Bayes risk, which is at least the minimum Bayes risk. This proves the lower bound for the full class. Identification follows because the law under the allowable experiment $g$ contains the distribution of $Y^g$ itself. ∎

Corollary (A finite-sample point-parameter confidence interval): Under this design, for $0<\alpha<1$, \[ I_n=\left[\overline Y-\frac1{2\sqrt{n\alpha}},\ \overline Y+\frac1{2\sqrt{n\alpha}}\right]\cap[0,1] \] has coverage at least $1-\alpha$ uniformly in the model class.

Proof: Independence gives ${\mathbb{E}}[(\overline Y-\theta)^2]\le1/(4n)$. Applying Markov's inequality to the nonnegative square gives the claim. Intersecting with the known parameter space cannot remove the true $\theta$. ∎

The shrinkage estimator is used for exact minimax risk; the unshrunk mean is used for this elementary confidence interval. They are not silently interchanged in either calculation.

Restricted support: exact nonidentification and risk

Theorem: Take $K=1$, no covariates, binary $W$, and a latent $U\sim\operatorname{Bernoulli}(\theta)$. Let $Y=WU$. Permit only $W=0$, and target $g=\operatorname{do}(W=1)$. Then the graph can be fixed, the identified set is exactly $[0,1]$, and $R_n=1/4$ for every $n\ge0$.

Proof: The structural equations define unique counterfactuals; the DAG has arrows $U\to Y$ and $W\to Y$. Under every allowable experiment $W=Y=0$, so every $\theta$ induces the same observed law. Under $g$, $Y^g=U$, hence its expectation can be any $\theta\in[0,1]$. There are no values outside this range because outcomes are bounded; the identified set is sharp.

The estimator $1/2$ has worst squared error $1/4$. Conversely, for any randomized estimator $D$ under the common data law, the average of its risks at $\theta=0$ and $\theta=1$ is \[ \tfrac12{\mathbb{E}}[D^2+(D-1)^2]={\mathbb{E}}[(D-1/2)^2]+\tfrac14\ge\tfrac14. \] The maximum of the two endpoint risks is at least their average. This proves equality. ∎

Proposition (Confidence-set obstruction): In Theorem Theorem, if a measurable random set $C_n\subseteq[0,1]$ has pointwise coverage at least $1-\alpha$ for every possible $\theta$, then ${\mathbb{E}}[\operatorname{Leb}(C_n)]\ge1-\alpha$. If it must cover the entire identified set with probability at least $1-\alpha$, it must equal $[0,1]$ up to endpoints on that event.

Proof: All parameters have the same data law $Q$, so $Q(\theta\in C_n)\ge1-\alpha$ for all $\theta$. Tonelli's theorem applies to the nonnegative measurable membership indicator and gives ${\mathbb{E}}_Q\operatorname{Leb}(C_n)=\int_0^1Q(\theta\in C_n)\,d\theta\ge1-\alpha$. The second assertion follows directly from the definition of simultaneous set coverage. ∎

Supported adaptive designs: a horizon-dependent upper bound

Theorem: Let the target be a fixed action path $w_{1:K}$. Suppose every stage assignment has known probability $q_{i,k}(w_k\mid H_{i,k})\ge\varepsilon>0$ on all histories reachable under the target, and uses the external randomization specified above. Define \[ L_i=\prod_{k=1}^K\frac{{\mathbf 1}\{W_{i,k}=w_k\}}{q_{i,k}(w_k\mid H_{i,k})}, \quad Z_i=L_iY_i,\quad\widehat\theta=n^{-1}\sum_i Z_i, \] with a factor set to zero once the target path has not been followed. Then \[ {\mathbb{E}}[Z_i\mid\mathcal F_{i-1}]=\theta,\qquad {\mathbb{E}}[(\widehat\theta-\theta)^2]\le\frac{\varepsilon^{-K}}n. \]

Proof: Condition on previous episodes and on all the fresh episode's structural errors. These errors specify the realized history along the target path and its outcome. The conditional probability that successive external assignment draws follow that path is the product of their target probabilities evaluated at those histories. Multiplying the path indicator by their reciprocal therefore integrates to one. With an outcome factor, it integrates to that error realization's $Y^g$. Integrate over the fresh errors, which are independent of $\mathcal F_{i-1}$, to obtain ${\mathbb{E}}[L_i\mid\mathcal F_{i-1}]=1$ and ${\mathbb{E}}[Z_i\mid\mathcal F_{i-1}]=\theta$. This is a direct sequential change-of-measure calculation, not an assumption of no latent confounding in the observational distribution.

On the event of following the target, $L_i\le\varepsilon^{-K}$ and $0\le Y_i\le1$, so $Z_i^2\le L_i^2\le\varepsilon^{-K}L_i$. Thus the conditional second moment is at most $\varepsilon^{-K}$. For $i<j$, the tower rule gives ${\mathbb{E}}[(Z_i-\theta)(Z_j-\theta)]=0$, since $Z_i-\theta$ is $\mathcal F_{j-1}$-measurable and the conditional mean of the other factor is zero. Summing variances gives the bound. The same argument with Markov's inequality yields a confidence radius $\sqrt{\varepsilon^{-K}/(n\alpha)}$, intersected with $[0,1]$. ∎

This bound need not be minimax sharp, and can exceed the trivial risk $1/4$. It supplies neither a universal exponential-in-$K$ lower bound nor a claim that inverse weighting is optimal.

VERIFICATION

For $n=1$, Theorem Theorem gives $(Y+1/2)/2$ and risk $1/16$ on Bernoulli observations; the formula gives $1/16$ as well. For $n=0$, a separate midpoint argument gives $1/4$; the beta-prior proof was deliberately stated only for $n\ge1$. In the nonidentification example, $n=1,2,3$ all produce exactly the same all-zero record, not merely close distributions.

\begin{lstlisting} for theta in {0, 1/2, 1}:     enumerate S = 0,...,n with binomial probabilities     compute mean squared error of (S + sqrt(n)/2)/(n+sqrt(n))     compare with 1/(4*(sqrt(n)+1)^2) # For the restricted-support model enumerate the two latent endpoints: # both return W=Y=0, while E[Y under do(W=1)] is 0 or 1. \end{lstlisting}

This is reproducible pseudocode for the closed-form finite checks; the proof uses exact identities, not an unreported large simulation. No algorithmic search over all causal graphs was run.

Adversarial checks: the minimax lower bound requires the stated Bernoulli submodel to lie in the structural class; a graph restriction making $\theta$ known would invalidate that lower bound. The direct-intervention upper bound requires actual permission to implement $g$, not merely a wish to randomize. The weighting calculation conditions on fresh structural errors only to integrate external randomization and then removes that conditioning; it does not replace a causal conditional distribution with an observational one. Across-episode independence of observed records was never assumed for adaptive designs.

FINAL

LABEL: UNRESOLVED. Fully proved subcases: exact full-control risk $1/[4(\sqrt n+1)^2]$ under the explicit richness condition; exact restricted-support identified set $[0,1]$ and risk $1/4$; and a supported adaptive weighting bound with valid confidence intervals.

Exact first gap. Given an arbitrary declared latent structural class and restricted intervention family, no necessary-and-sufficient usable criterion for constancy of $\theta$ on experimental equivalence classes is derived here.

Three next moves. (1) Solve that criterion for finite latent-state models with a specified subset of manipulable nodes. (2) Derive testing lower bounds for pairs agreeing on accessible interventions but separated under $g$, tracking the smallest target-path support. (3) Optimize design over the resulting identified functional and prove matching risk and confidence-set bounds rather than assuming inverse weighting is minimax.

Minimal hard model. One binary unmanipulable treatment coordinate, a latent parent, and a target counterfactual outside experimental support already create the first obstruction. Approximate rather than exact equivalence is the likely source of harder positive-identification rates.

Completion Estimate

30%

Two sharp opposite regimes and a general supported-design bound are proved; the requested classification over arbitrary restricted families and matching minimax bounds remain open within this attempt.

This is a subjective estimate of the fraction of the whole requested mathematical scope addressed by this report, including the named variants. It is not a probability of correctness, an objectively measured fraction of a proof, or an estimate that the remaining work takes the complementary fraction of the time. Only the eleven supplied files are assessed; no missing rank-1--200 statements are inferred.

References

[causal] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference. arXiv:2508.17099v1 (2025), Section 3.1.3. https://arxiv.org/html/2508.17099v1.

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Econ-43-A1 · Exploration · LLM-generated · gpt 6 astra ultra · 2026-10-09

GPT 6 Astra Ultra attempt on Econ-43

Sequential adaptive experiments with uncertain causal structure Support, martingale estimation, and an adaptive lower bound

Problem ID: C90-42. Status: Unresolved; rigorous restricted results.

Short recap and scope

The target combines identification and minimax estimation when sequential interventions are restricted and the causal graph is uncertain. The results here give matching risk bounds in a fully manipulated subfamily, show why adaptation cannot repair forbidden interventions, and give a sharp partial-identification interval for a restricted one-stage design. The broad problem over arbitrary graph families remains unresolved.

Cinelli et al. [challenge], Section 3.1.3, motivates increased adaptivity and sequential treatments. The weighting identity below is the usual sequential randomization argument, proved here for adaptive episodes; it does not assume that episodes remain independent after design choices.

A fully manipulated regime

Fix $K\geq1$. In episode $i$, a fresh structural-error vector is independent of previous episodes and of the external assignment randomizers. Its law is the same across episodes. Let $H_{ik}$ be the observed history before the $k$th action and $\mathcal F_{i-1}$ all previous data and design randomness. A fixed deterministic regime $g$ prescribes $g_k(H_{ik})$. Assignments have known probabilities $q_{ik}(w\mid H_{ik},\mathcal F_{i-1})$ and replace the structural equations for every treatment coordinate. Conditional on the observed history, their randomizers are independent of structural errors. Assume consistency, unique counterfactual trajectories, $0\leq Y_i\leq1$, and \[ q_{ik}(g_k(H_{ik})\mid H_{ik},\mathcal F_{i-1})\geq\eta>0 \] at every regime-relevant history. Put $\theta={\mathbb{E}} Y^g$ and \[ Z_i=Y_i\prod_{k=1}^K \frac{{\mathbf 1}\{W_{ik}=g_k(H_{ik})\}} {q_{ik}(g_k(H_{ik})\mid H_{ik},\mathcal F_{i-1})}, \qquad \widehat\theta=n^{-1}\sum_i Z_i. \]

Theorem (Identification and adaptive-episode risk): Under these assumptions ${\mathbb{E}}[Z_i\mid\mathcal F_{i-1}]=\theta$ and \[ {\mathbb{E}}(\widehat\theta-\theta)^2 \leq \frac{\eta^{-K}\theta-\theta^2}{n} \leq\frac{\eta^{-K}}n. \] Clipping $\widehat\theta$ to $[0,1]$ cannot increase its risk. Consequently the observable law of any such known design identifies $\theta$, regardless of latent confounding among the variables whose treatment equations have all been replaced.

Proof: Condition on the past and the fresh structural-error vector. On the event that all assignments follow $g$, consistency determines the whole trajectory and $Y_i=Y_i^g$. For that vector the probability of this event is the product $\pi_i^g$ of the assignment probabilities evaluated along the regime trajectory: integrate the independent stage randomizers in chronological order. The weight on that event is $1/\pi_i^g$. Thus its conditional weighted expectation is $Y_i^g$. Averaging over the fresh errors gives $\theta$, independently of the past. This same argument is valid with latent dependent structural components, since only assignment randomizers must be external. Let $R_i$ be the weight product. Then $0\leq R_i\leq\eta^{-K}$ and $Z_i^2=R_i^2Y_i^2\leq\eta^{-K}R_iY_i$. Hence ${\mathbb{E}}[Z_i^2\mid\mathcal F_{i-1}]\leq\eta^{-K}\theta$. The centered variables are martingale differences: for $i<j$, ${\mathbb{E}}[(Z_i-\theta)(Z_j-\theta)]=0$ by conditioning on $\mathcal F_{j-1}$. Summing their variances proves the risk bound. Projection onto $[0,1]$ weakly reduces distance to every $\theta\in[0,1]$. Finally the expectation of the observable, known-weight random variable $Z_i$ is determined by its data law, proving identification. ∎

For example Chebyshev's inequality immediately yields the conservative confidence interval $[\widehat\theta-\sqrt{\eta^{-K}/(n\alpha)}, \widehat\theta+\sqrt{\eta^{-K}/(n\alpha)}]\cap[0,1]$ with coverage at least $1-\alpha$. This shrinks for fixed $K,\eta$. Clipping the center first also preserves the coverage assertion.

Matching horizon cost when target assignments are capped

Consider binary actions, no intermediate covariates, regime $g=(1,\ldots,1)$, and all allowable policies satisfying $q_{ik}(1\mid H_{ik},\mathcal F_{i-1})\leq r$, $0<r\leq1$. Suppose the constant policy with probability $r$ at every stage is allowable. The model class contains structural models \[ Y_i={\mathbf 1}\{W_{i1}=\cdots=W_{iK}=1\}{\mathbf 1}\{U_i\leq\theta\}, \quad U_i\text{ independent uniform on }[0,1],\quad \theta\in[1/4,3/4]. \] These models are a subfamily of an admissible DAG with arrows from every action to $Y$; no unknown graph is needed for the lower bound.

Lemma (Two elementary testing inequalities): For distributions $P,Q$ with finite ${\operatorname{KL}}(P\|Q)$, ${\operatorname{TV}}(P,Q)\leq\sqrt{{\operatorname{KL}}(P\|Q)/2}$. If target values differ by $\Delta$, every estimator has maximum squared-error risk at least $\Delta^2[1-{\operatorname{TV}}(P,Q)]/8$. For Bernoulli probabilities $u,v\in(0,1)$, \[ {\operatorname{KL}}(\operatorname{Ber}(u)\|\operatorname{Ber}(v)) \leq (u-v)^2/[v(1-v)]. \]

Proof: For an event $A$, grouping the likelihood into $A,A^c$ and applying Jensen's inequality for $t\log t$ shows ${\operatorname{KL}}(P\|Q)\geq\operatorname{kl}(P(A),Q(A))$. As a function of its first argument $x$, binary relative entropy has second derivative $1/[x(1-x)]\geq4$, and value and first derivative zero at $x=Q(A)$. Integration twice gives $\operatorname{kl}(P(A),Q(A))\geq2(P(A)-Q(A))^2$. Taking the supremum over $A$ proves the first inequality, including limits at the endpoints. For the second, classify an estimate by which target is nearer. On a misclassification its squared error is at least $\Delta^2/4$. The sum of the two error probabilities is at least $1-{\operatorname{TV}}(P,Q)$; the maximum risk is at least half their risk sum. Finally $\log x\leq x-1$ in the two terms of binary entropy gives $u(u/v-1)+(1-u)((1-u)/(1-v)-1)=(u-v)^2/[v(1-v)]$. ∎

Theorem (Sharp risk order for the capped-design subfamily): There is a numerical $c>0$ such that for every $n,K,r$, every allowable adaptive design and every estimator, \[ \sup_{\theta\in[1/4,3/4]}{\mathbb{E}}_\theta(\widehat\theta-\theta)^2 \geq c\min\{1,(nr^K)^{-1}\}. \] For the larger class of all bounded structural responses under fully manipulated actions, the nonadaptive probability-$r$ design and clipped weighting estimator have risk at most $\min\{1,(nr^K)^{-1}\}$. Thus the order is matched for any class between these two specified classes.

Proof: Let $\delta=\frac1{16}\min\{1,(nr^K)^{-1/2}\}$ and $\theta_-=1/2-\delta,\theta_+=1/2+\delta$. Use the same randomized algorithm under both models. In the chain rule for the relative entropy of its transcript, its design kernels and internal randomizers have the same conditional law given the same past, so contribute zero. An outcome contributes no information unless all $K$ actions equal one. Conditional on the past this event has probability at most $r^K$, by multiplying the stage bounds. On that event its entropy is $\operatorname{kl}(\theta_-,\theta_+)\leq32\delta^2$, using the lemma and $\theta_+(1-\theta_+)\geq1/8$. Therefore transcript entropy is at most $32nr^K\delta^2\leq1/8$. The lemma gives total variation at most $1/4$, and risk at least $(2\delta)^2(3/4)/8=3\delta^2/8$, which has the claimed order. For the upper bound the preceding theorem has $\eta=r$, and clipping also bounds squared error by one. The proof needs no exploration benefit assumption and includes all adaptive history-based choices. ∎

The exponential factor is a property of this sequential cap. If the whole target sequence can be assigned deterministically, $r=1$ and it disappears. It is therefore not a universal lower bound for all experimental designs.

Forbidden actions: a sharp identified interval

Consider $K=1$, observed $X$ in a standard Borel space, and a known measurable subset $B$. Its distribution $P_X$ is observed. Treatment can be randomized with positive probabilities for both levels on $B$, but must equal zero on $B^c$. Responses have unrestricted joint potential outcome distributions conditional on $X$, with $0\leq Y(w)\leq1$. Assignments are external and there is no interference. Let $a=\int_B {\mathbb{E}}[Y(1)\mid X=x]\,dP_X(x)$ and $b=P_X(B^c)$.

Theorem (Sharp support obstruction): The family of all allowable adaptive experiment laws identifies $a,b$ and has sharp identified set $[a,a+b]$ for ${\mathbb{E}} Y(1)$. If $b>0$, every estimator under every allowable design has maximum risk over the observational equivalence class at least $b^2/4$, for every sample size.

Proof: On $B$, external assignments with positive support identify the treated conditional mean, and $X$ identifies its integration measure. On $B^c$ the unobserved treated mean lies in $[0,1]$, yielding the interval. For any $t\in[0,1]$, retain all observed conditional response laws and replace $Y(1)$ on $B^c$ by a Bernoulli variable of conditional mean $t$, independent of the remaining potential outcomes given $X$. Such a joint law exists by a product construction. It respects consistency and external randomization and changes no allowable observable law, even adaptively, because treatment there is always zero. Its target is $a+bt$, proving sharpness. For the endpoint models the estimator has a common distribution, say that of $T$. Pointwise, $[(T-a)^2+(T-a-b)^2]/2=(T-a-b/2)^2+b^2/4$. Expectation and the maximum-over-endpoints bound prove the last claim. ∎

Verification and remaining gap

The weighting variance uses $\eta^{-K}$, not the looser squared bound $\eta^{-2K}$; this follows from nonnegativity and the weighted expectation identity. The lower bound compares complete adaptive transcripts. The support interval uses an unrestricted potential-outcome class and need not remain sharp after additional graph restrictions. The first unresolved step is to characterize which partially manipulable, graph-uncertain classes allow cross-regime constraints to shrink that interval and then to derive their design-dependent minimax rates. No general causal-graph identification algorithm is claimed.

References

[challenge] 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 3.1.3, Sequential treatments and increased adaptivity. https://arxiv.org/html/2508.17099v1.

Completion Estimate

25%

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Record details

Subjects: Economics
Collection: Economics catalogue
Economics rank: 43
JEL: C90 — General