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Econ-263 · economics · Economics

Confidence sequences for adaptively evolving causal targets

LLM status Open unverified

Statement

CI-02 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Let $\mathcal F_t$ contain the observations and design randomness through time $t$. At stage $t$, an $\mathcal F_{t-1}$-measurable regime $g_t$ is chosen from a finite treatment set. Potential outcomes satisfy $Y_t(w)\in[0,1]$, and the assignment $W_t$ is conditionally randomized with known probabilities $q_t(w)\geq\varepsilon>0$, independently of the current potential outcomes given $\mathcal F_{t-1}$. Consistency gives $Y_t=Y_t(W_t)$; between-unit interference is excluded in this particular model. The potential-outcome distribution may drift with the past.

A well-defined moving target is the cumulative predictable mean \[ \theta_t(P)=\frac1t\sum_{s=1}^t \E_P[Y_s(g_s)\mid\mathcal F_{s-1}], \] which is random because the selected regimes and histories are random. A confidence sequence is a sequence of $\mathcal F_t$-measurable intervals $C_t$.

Formal question

Characterize predictable target families and drift restrictions for which \[ \inf_{P\in\mathcal M}\Pr_P\{\theta_t(P)\in C_t \text{ for every }t\geq1\}\geq1-\alpha \] can hold together with a uniformly shrinking, explicitly quantified interval width. Determine optimal width orders as a function of the assignment probabilities, time, and target variation, and prove matching lower bounds. Separately characterize when an instantaneous target $\E_P[Y_t(g_t)\mid\mathcal F_{t-1}]$ admits shrinking intervals: unconstrained temporal drift must not be replaced silently by stationarity.

Scope and status

This formalization distinguishes predictable adaptation from choosing a target after inspecting its own outcome. Coverage automatically includes stopping at any almost surely finite stopping time. Standard fixed-target randomized examples already have confidence-sequence methods; their general existence is not presented as a new conjecture. The attached agenda concerns extensions and sharp limits for changing causal targets.

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 1

Econ-263-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-263

Anytime inference for predictable moving causal targets

Problem ID: C13-259. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

Question and target

The sequential-causal agenda [Causal] asks for inference when both assignment and the target policy evolve. Time-uniform inference is already established in the literature [CS]; existence in a fixed-target model is not new. We supply an explicit confidence sequence for the predictable running target under arbitrary drift, a lower bound for its information scale, and an obstruction for instantaneous targets.

At date $t$, the finite-action rule $g_t$ and assignment probabilities $q_t(a)$ are measurable with respect to the past $\mathcal F_{t-1}$. Conditional on the past, treatment $W_t$ is randomized independently of the vector of potential outcomes, with known $q_t(a)\ge\varepsilon>0$. Outcomes lie in $[0,1]$ and consistency gives $Y_t=Y_t(W_t)$. There is no stationarity assumption. Define \[ \mu_t={\mathbb{E}}[Y_t(g_t)\mid\mathcal F_{t-1}],\qquad \theta_t=\frac1t\sum_{s=1}^t\mu_s, \] \[ Z_t=\frac{{\mathbf 1}\{W_t=g_t\}Y_t}{q_t(g_t)},\quad D_t=Z_t-\mu_t,\quad V_t=\sum_{s=1}^t\frac1{q_s(g_s)},\quad R=1/\varepsilon. \] Sequential randomization gives ${\mathbb{E}}[D_t\mid\mathcal F_{t-1}]=0$, $|D_t|\le R$, and conditional second moment at most $1/q_t(g_t)$. Notice that $\theta_t$ is random and predictable in its summands. The martingale identity remains valid for this target.

An explicit stitched confidence sequence

Fix $\eta>1$ and $\alpha\in(0,1)$. For integers $k\ge0$, put \[ \alpha_k=\frac{6\alpha}{\pi^2(k+1)^2},\quad x_k=\log(2/\alpha_k),\quad v_k=\eta^{k+1},\quad b_k=\sqrt{2v_kx_k}+\frac R3x_k. \] Since $V_t\ge t\ge1$, the index $k(t)=\lfloor\log_\eta V_t\rfloor$ is nonnegative. Let $\widetilde\theta_t$ be the projection of $t^{-1}\sum Z_s$ onto $[0,1]$ and define \[ C_t=\left[\widetilde\theta_t-\frac{b_{k(t)}}t, \widetilde\theta_t+\frac{b_{k(t)}}t\right]\cap[0,1]. \]
Theorem (Uniform coverage with unrestricted predictable drift): Under the stated assumptions, \[ {\mathbb{P}}\{\theta_t\in C_t\text{ for every integer }t\ge1\}\ge1-\alpha. \] The interval is nonempty and has diameter at most \[ \min\left\{1,\frac{2\sqrt{2\eta V_t x_{k(t)}}}{t} +\frac{2Rx_{k(t)}}{3t}\right\}. \] In particular its deterministic upper bound is of order \[ \sqrt{\frac{\log\log(et/\varepsilon)+\log(1/\alpha)}{\varepsilon t}} +\frac{\log\log(et/\varepsilon)+\log(1/\alpha)}{\varepsilon t}, \] with constants depending on $\eta$. Coverage holds at every almost surely finite stopping time.
Proof: For $0<\lambda<3/R$, expand the conditional exponential moment and use ${\mathbb{E}}[D_t\mid\mathcal F_{t-1}]=0$, $|D_t|^m\le R^{m-2}D_t^2$, and $m!\ge2\,3^{m-2}$ to obtain \[ {\mathbb{E}}[e^{\lambda D_t}\mid\mathcal F_{t-1}] \le\exp\left\{\frac{\lambda^2}{2(1-\lambda R/3)q_t(g_t)}\right\}. \] The same bound holds for $-D_t$. Thus $M_t=\exp\{\lambda\sum_{s\le t}D_s-\lambda^2V_t/[2(1-\lambda R/3)]\}$ is a nonnegative supermartingale starting at one. For any $x>0$, stop it at its first crossing of $e^x$ and at a finite horizon. Conditional expectation iteration gives expectation at most one, hence crossing probability at most $e^{-x}$; let the horizon tend to infinity.

For epoch $k$ choose \[ \lambda_k=\frac{\sqrt{2x_k/v_k}}{1+(R/3)\sqrt{2x_k/v_k}}. \] If $V_t<v_k$ and $\sum_{s\le t}D_s\ge b_k$, substitution shows $\lambda_k\sum D_s-\lambda_k^2V_t/[2(1-\lambda_kR/3)]\ge x_k$. The supermartingale inequality therefore bounds an upper crossing in epoch $k$ by $\alpha_k/2$, and likewise a lower crossing. Summing over $k$ gives total probability at most $\sum_k\alpha_k=\alpha$. On the complementary event, $|t^{-1}\sum Z_s-\theta_t|\le b_{k(t)}/t$ at every date. Projection onto $[0,1]$ cannot increase the error, proving coverage and nonemptiness. Since $v_{k(t)}\le\eta V_t$ and $t\le V_t\le t/\varepsilon$, the diameter and rate follow. A simultaneous event includes every finite stopping time without further optional-stopping assumptions. ∎

Information limits

Proposition (A fixed-time lower bound): For every $t$, over the stated model class with two treatments and $0<\varepsilon\le1/2$, the minimax mean squared error for $\theta_t$ is at least $c\min\{1,1/(\varepsilon t)\}$ for a universal $c>0$. For any fixed $\alpha<1/4$, every confidence interval honest over this class at date $t$ has expected diameter at least $c_\alpha\min\{1,1/\sqrt{\varepsilon t}\}$ under some law.
Proof: Restrict to $g_s=1$, independent assignment with probability $\varepsilon$, $Y_s(0)=0$, and independent $Y_s(1)\sim\operatorname{Bernoulli}(\theta)$. The target is the constant $\theta$. Compare $\theta_0=1/2$ and $\theta_1=1/2+\delta$, $0<\delta\le1/4$. The divergence between their observed laws is at most $4t\varepsilon\delta^2$, since only observations assigned treatment one carry information. Choose $\delta=c_0\min\{1,1/\sqrt{t\varepsilon}\}$ small enough that total variation is less than a fixed $v<1-2\alpha$, using ${\operatorname{TV}}\le\sqrt{{\operatorname{KL}}/2}$. Every test has sum of errors at least $1-v$. Decoding which of the two target values is closest to an estimate gives the squared-error lower bound. If $I$ has coverage $1-\alpha$ under both laws, then under law zero, \[ {\mathbb{P}}_0\{\theta_0,\theta_1\in I\}\ge1-2\alpha-v>0, \] because the probability of covering $\theta_1$ changes by at most total variation. Its diameter is at least $\delta$ on this event, proving the second assertion. The elementary variation-divergence inequality follows by binary aggregation and convexity of binary relative entropy. ∎
This establishes the effective information scale $t\varepsilon$ at a fixed date. It does not prove the sharp iterated-logarithm constant or a full time-uniform minimax lower bound.
Proposition (Arbitrary drift prevents shrinking instantaneous inference): Fix $\alpha<1/4$. For every date $t$, any interval with coverage at least $1-\alpha$ for the instantaneous mean $\mu_t$ under every unrestricted-drift law has expected diameter at least $(1/2)(1/2-2\alpha)$ under some law, even if the target outcome is observed every round.
Proof: Use a single-action experiment. Construct two laws identical through $t-1$, with the current outcome Bernoulli of mean $1/4$ under the first and $3/4$ under the second. Their observations through $t$ have total variation $1/2$. Under the first law the event that the interval covers both current means has probability at least $1-2\alpha-1/2$. On that event its diameter is at least $1/2$. Past observations cannot help distinguish these two current-date alternatives. ∎

What a drift restriction buys

For illustration suppose $|\mu_s-\mu_{s-1}|\le L$ almost surely with known $L$. For every $1\le h\le t$, a window estimator $\bar Z_{t,h}=h^{-1}\sum_{s=t-h+1}^t Z_s$ targets the window average, whose distance from $\mu_t$ is at most $L(h-1)/2$. Let \[ x_{t,h}=\log\left(\frac{\pi^4t^2h^2}{18\alpha}\right). \] A fixed-window martingale Bernstein bound with conditional variance at most $h/\varepsilon$, followed by a union bound, gives simultaneous instantaneous coverage using radii \[ \sqrt{\frac{2x_{t,h}}{\varepsilon h}}+ \frac{2x_{t,h}}{3\varepsilon h}+\frac{L(h-1)}2. \] Indeed each two-sided error probability is at most $2e^{-x_{t,h}}$; their sum over $t\ge1,h\le t$ is at most $36\alpha\pi^{-4}(\sum t^{-2})^2=\alpha$. The bias bound is the sum of $0,L,\ldots,(h-1)L$ divided by $h$. Thus even a data-chosen window retains coverage on the simultaneous event. The variance-bias tradeoff suggests $h$ of cubic-root order in $1/(\varepsilon L^2)$ up to logarithms, rather than a shrinking interval under fixed unrestricted drift. Adaptive unknown drift, estimated propensities, and sharp efficiency for richer evolving causal targets remain unresolved here.

Completion Estimate

60%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt constructs shrinking confidence sequences for predictable cumulative causal means under arbitrary drift, proves fixed-time information bounds and rules out shrinking instantaneous intervals without drift restrictions. Matching time-uniform lower bounds, sharp width orders and a broader characterization of target families and adaptive drift restrictions remain incomplete.

References

  • [Causal] 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.
  • [CS] S. R. Howard, A. Ramdas, J. McAuliffe and J. Sekhon (2021), "Time-uniform, nonparametric, nonasymptotic confidence sequences," Annals of Statistics 49(2), 1055--1080. arXiv:1810.08240. The supermartingale and stitching method is established methodology; the target and limitations here are specified separately.

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