Model-X anytime-valid conditional independence testing via e-processes
Mathematical Methods AIMech:ModelXAnytimeValidConditionalIndependenceTesting DRAFT
Aliases: Model-X Conditional Independence E-Statistics, Sequential E-CRT (e-Conditional Randomization Test), Monte Carlo Plus-One Model-X E-Process, Conditional GRO E-Variable for Model-X Testing
A sequential conditional independence testing framework (Grünwald, Henzi, & Lardy 2024, Journal of the American Statistical Association 119(546):1554-1565, arXiv:2209.12637; Shaer, Maman, & Romano 2023, NeurIPS 36, arXiv:2210.00354) for testing H_0: X _||_ Y | Z under the Model-X assumption (where the conditional law Q_{X|Z} is known or estimated while the outcome conditional law P_{Y|X,Z} is completely unspecified) that constructs conditional e-variables E(X_n; Y_n, Z_n) = f(X_n, Y_n, Z_n) / int f(x, Y_n, Z_n) dQ_{X|Z=Z_n}(x) (or exact finite-sample Monte Carlo plus-one approximations E^{(M)}(X_n; Y_n, Z_n) = f(X_n, Y_n, Z_n) / ((1/(M+1))(f(X_n, Y_n, Z_n) + sum_{i=1}^M f(tilde{X}_n^{(i)}, Y_n, Z_n))) using M >= 1 conditional randomization draws tilde{X}_n^{(i)} ~ Q_{X|Z=Z_n}) whose product M_n = prod_{i=1}^n E_i forms a non-negative test martingale under H_0 regardless of the serial dependence of (Y_n, Z_n), achieves Growth-Rate Optimality (GRO) equal to the conditional mutual information I(X; Y | Z) via a conditional Kullback-Leibler divergence regret decomposition, and admits nonasymptotic Total Variation type-I error inflation bounds under Model-X misspecification.
Identity Notes & Concept Scope
No normalized identity match for Model-X anytime-valid conditional independence testing via e-processes across all 442 pinned AIO records and 582 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only prior discovery frontier entries where Grünwald, Henzi, & Lardy (2024, JASA 119(546):1554-1565, arXiv:2209.12637) and Shaer, Maman, & Romano (2023, NeurIPS 36, arXiv:2210.00354) were tracked as a pending frontier lead. Distinct from AIMech:SequentialKernelizedIndependenceTestingByBetting (Podkopaev et al. 2023), which tests unconditional independence H_0: X _||_ Y without knowing either marginal by pairing adjacent observations into an i.i.d. exchangeability stream; distinct from AIMech:KernelConditionalIndependenceTest (Zhang et al. 2011) and AIMech:GeneralisedCovarianceMeasureForConditionalIndependence (Shah & Peters 2020), which are fixed-sample asymptotic conditional independence tests without anytime-valid optional-stopping control; and distinct from AIMech:ModelXKnockoffsForHighDimensionalControlledVariableSelection (Candès et al. 2018) and AIMech:DerandomizedKnockoffsWithEValues (Ren & Barber 2024), which perform batch false discovery rate variable selection rather than sequential anytime-valid conditional independence testing via predictable e-processes.
Claims & Grounded Evidence
4 Claims
Mechanism
#conditional-e-variable-monte-carlo-plus-one-exchangeability-and-test-martingale
For any non-negative measurable test function f(x, y, z) >= 0 with 0 < int f(x, y, z) dQ_{X|Z=z}(x) < infty, the normalized ratio E(X; Y, Z) = f(X, Y, Z) / int f(x, Y, Z) dQ_{X|Z=z}(x) satisfies E_P[E(X; Y, Z) | Y, Z] = 1 for every joint distribution P in H_0: X _||_ Y | Z with conditional law Q_{X|Z} (Proposition 1, Equation 2); when the integral denominator is intractable, drawing M >= 1 conditional randomization copies tilde{X}^{(1)}, ..., tilde{X}^{(M)} ~_{i.i.d.} Q_{X|Z} conditionally independent of (X, Y) given Z and including the observed X in the denominator yields the Monte Carlo plus-one e-variable E^{(M)}(X; Y, Z) = f(X, Y, Z) / ((1/(M+1))(f(X, Y, Z) + sum_{i=1}^M f(tilde{X}^{(i)}, Y, Z))), which satisfies E_P[E^{(M)}(X; Y, Z) | Y, Z] = 1 exactly by conditional exchangeability of (X, tilde{X}^{(1)}, ..., tilde{X}^{(M)}) given (Y, Z) while the naive Monte Carlo estimator without the observed X in the denominator violates E[E] <= 1 (Proposition 2, Equation 3); sequentially multiplying E_n with predictable functions f_n measurable wrt sigma(X^{(n-1)}, Y^{(n)}, Z^{(n)}) forms a non-negative test martingale M^{(n)} = prod_{i=1}^n E_i with E_P[M^{(n)}] = 1 even under arbitrary temporal dependence of (Y_n, Z_n), provided X_n _||_ (X^{(n-1)}, Y^{(n)}, Z^{(n-1)}) | Z_n (Proposition 3, Equation 4).
Mechanism
#gro-conditional-mutual-information-and-conditional-kl-regret-identity
Under a point alternative P_1 = P_{Z,Y} x P_{X|Y,Z} dominated by H_0 with positive densities p(x | y, z) and q(x | z), the Growth-Rate Optimal (GRO) conditional e-variable maximizing E_{P_1}[log E(X; Y, Z) | Y, Z] is uniquely E^{GRO}(X; Y, Z) = p(X | Y, Z) / q(X | Z), whose expected log-growth rate under P_1 equals the conditional mutual information I(X; Y | Z) = E_{P_1}[D_KL(P_{X|Y,Z} || Q_{X|Z})] >= 0 (Section 3.1, Theorem 4, Equations 5–6); furthermore, for any non-negative test function f(x, y, z) defining the tilted conditional density r_f(x | y, z) = f(x, y, z) q(x | z) / int f(x', y, z) q(x' | z) dx', the conditional expected log-growth satisfies the exact identity E_{X~P_{X|Y=y,Z=z}}[log E_f(X; y, z)] = D_KL(P_{X|Y=y,Z=z} || Q_{X|Z=z}) - D_KL(P_{X|Y=y,Z=z} || R_{f,X|Y=y,Z=z}) (Section 3.1, Proposition 4, Equation 7), and when P_{Y|X,Z} belongs to a natural exponential family with density exp(theta^T T(X, Z) Y - A(theta | X, Z)), the plug-in test function f_hat(x, y, z) = p_{hat{theta}}(y | x, z) matches the oracle conditional density ratio up to estimation error in hat{theta} because baseline factors depending only on (y, z) cancel identically between numerator and denominator (Section 3.2, Equations 8–9).
Mechanism
#total-variation-type-i-error-bound-under-model-x-misspecification
When the true covariate law Q^*_{X^{(n)}|Z^{(n)}} differs from the assumed or estimated law hat{Q}_{X^{(n)}|Z^{(n)}} used to normalize the e-variables hat{E}_i, if hat{E}_i is uniformly bounded by B > 0 then E_{Q^*}[hat{E}_i | Y_i, Z_i] <= 1 + B * d_TV(Q^*_{X_i|Z_i}, hat{Q}_{X_i|Z_i}) (Proposition 5); crucially, for the stopped product process hat{M}_tau = prod_{i=1}^tau hat{E}_i at any stopping time tau <= n and level alpha in (0, 1), coupling Q^* and hat{Q} prior to Ville's maximal inequality proves that the anytime type-I error under the true null distribution satisfies P_{Q^*}(exists k <= n : hat{M}_k >= 1/alpha | Y^{(n)}, Z^{(n)}) <= alpha + d_TV(Q^*_{X^{(n)}|Z^{(n)}}, hat{Q}_{X^{(n)}|Z^{(n)}}) <= alpha + sqrt((1/2) sum_{i=1}^n D_KL(Q^*_{X_i|Z_i} || hat{Q}_{X_i|Z_i})) without requiring any boundedness assumption on hat{E}_i (Section 4, Theorem 6).
Mechanism
#truncation-protection-against-zero-lock-in-and-sequential-e-crt-betting
Because a single outlier step with f_i(X_i, Y_i, Z_i) approx 0 can drive an untruncated product martingale close to zero and permanently stall evidence accumulation during burn-in, mixing or lower-truncating the raw conditional e-variable via E_{i,eps} = (1 - eps) E_i + eps or f_{i,eps}(x, y, z) = max(f_i(x, y, z), eps * int f_i(x', y, z) dQ_{X|Z=z}(x')) with eps in (0, 1) (e.g., eps = 0.05) preserves exact conditional expectation E_{H_0}[E_{i,eps} | Y_i, Z_i] = 1 while lower-bounding every single-step factor by eps (Section 3.2, Section 5, Equation 10); similarly, the sequential Model-X e-CRT (Shaer, Maman, & Romano 2023, NeurIPS 36, arXiv:2210.00354) constructs sequential betting e-variables E_t = 1 + lambda_t (s_t(X_t, Y_t, Z_t) - E_{Q_{X|Z}}[s_t(tilde{X}_t, Y_t, Z_t) | Y_t, Z_t]) with predictable bounded betting fractions lambda_t and arbitrary online black-box conformity scores s_t.
⚠️ Curated Limitations & Negative Results
- Valid Type-I error control relies on the Model-X assumption that the conditional distribution Q_{X|Z} is known or estimated with vanishing cumulative Total Variation / Kullback-Leibler error over the horizon n; when hat{Q}_{X_i|Z_i} has a persistent non-vanishing estimation bias d_TV(Q^*_{X_i|Z_i}, hat{Q}_{X_i|Z_i}) >= delta > 0 at every step, the product TV distance d_TV(Q^*_{X^{(n)}|Z^{(n)}}, hat{Q}_{X^{(n)}|Z^{(n)}}) approaches 1 as n -> infty, rendering the Theorem 6 robustness bound vacuous over long horizons (Section 4, p. 1559).
- When the conditional normalization integral int f(x, Y_n, Z_n) dQ_{X|Z=Z_n}(x) is approximated by the Monte Carlo plus-one estimator E^{(M)} (Proposition 2), each step is deterministically upper-bounded by M + 1, so achieving near-optimal GRO logarithmic growth under strong alternatives requires drawing M >> 1 conditional samples per time step (Section 2.1, p. 1556).
- Sequential plug-in estimation of the outcome model p_{hat{theta}_{n-1}}(y | x, z) from past samples requires O(n) cumulative refits (or online updates) and suffers low growth during initial burn-in when sample sizes are small relative to covariate dimension, requiring explicit truncation eps in (0, 1) to prevent early mispredictions from driving the product martingale near zero (Section 3.2, Section 5).
- All 636 lines of Grünwald, Henzi, & Lardy (2024, JASA 119(546):1554-1565, arXiv:2209.12637v2) were read. Constructed numerical checks in research/discovery/model-x-anytime-valid-conditional-independence-testing-boundaries.py verify (1) Section 2.1 Propositions 1-3 exact conditional unit expectation E_{H_0}[E | Y, Z] = 1.0 and Monte Carlo plus-one exact unit expectation E_{H_0}[E^{(M)} | Y, Z] = 1.0 (with max bound 2.333333 <= M+1 = 3 for M=2 and null anytime rejection rate 0.034 <= 0.10) vs. naive MC inflation E_{H_0}[E^{naive} | Y, Z] = 1.239524 > 1, (2) Section 3.1 Theorem 4 & Proposition 4 GRO optimality E_{P_1}[log E^{GRO}] = I(X; Y | Z) = 0.12312 > E_{P_1}[log E_f] = 0.096722 and exact conditional KL regret decomposition diff 0.0, (3) Section 4 Theorem 6 product TV robustness bound P_{Q^*}(exists k <= n : hat{M}_k >= 1/alpha) = 0.0 <= alpha + d_TV = 0.157797, and (4) Section 3.2 & Section 5 truncation protection (eps=0.05 floor 0.1 vs. untruncated 1.9e-13 under an outlier) and sequential logistic Model-X e-process null anytime Type-I error 0.025 <= 0.05 and alternative anytime power 1.0.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | Model-X conditional independence test martingales guarantee anytime-valid Type-I error control P_{H_0}(exists n >= 1 : M^{(n)} >= 1/alpha) <= alpha under arbitrary data-dependent optional stopping and continuous monitoring (Grünwald et al. 2024, Proposition 3), preventing optional-stopping false positives (AIO:DataDredgingBias). |
AIO:TimeSeriesAnalysis
Time Series Analysis
|
related_match | Proposition 3 establishes that the conditional e-process M^{(n)} = prod_{i=1}^n E_i remains a valid test martingale under arbitrary temporal dependence of the outcome and covariate streams (Y_n, Z_n)_{n>=1}, enabling sequential conditional independence testing in time-series and online streaming settings (AIO:TimeSeriesAnalysis). |
Evidence Sources & Primary Literature
4 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
grunwald2024-model-x-ci-ar5iv
|
Peter Grünwald, Alexander Henzi, Tyron Lardy
|
primary paper
full_text
|
ID: https://doi.org/10.1080/01621459.2023.2205607
sha256:2a68c4355f73…
|
grunwald2024-model-x-ci-crossref
|
Peter Grünwald, Alexander Henzi, Tyron Lardy
|
primary paper
abstract_only
|
ID: https://doi.org/10.1080/01621459.2023.2205607
sha256:8efce92ca7a7…
|
grunwald2024-model-x-ci-abs
|
Peter Grünwald, Alexander Henzi, Tyron Lardy
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.2209.12637
sha256:2ffb4cdf23e8…
|
shaer2023-ecrt-abs
|
Shalev Shaer, Gal Maman, Yaniv Romano
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.2210.00354
sha256:397b2ffe9dac…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T13:55:00Z
Updated: 2026-10-17T13:55:00Z
Source File:
data/curated/ModelXAnytimeValidConditionalIndependenceTesting.yaml
· Schema:
curated_concept.yaml
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