Post-Hoc Valid Inference

• Post-hoc valid inference is a family of frameworks that maintain rigorous error control after data-driven analysis, ensuring reliable significance tests and confidence intervals.
• Methodological paradigms include conditioning on selection events, constructing simultaneous guarantees, and employing orthogonal-score techniques to mitigate selection bias.
• The framework applies broadly to regression, multiple testing, causal discovery, and predictive uncertainty, effectively adapting to diverse, data-dependent decisions.

Searching arXiv for papers on post-hoc valid inference and closely related selective inference frameworks. Post-hoc valid inference denotes a family of inferential frameworks that preserve formal error control after a data-dependent analytical action has already been taken. The action may be variable selection, model selection, graph discovery, threshold exploration, adaptive choice of a significance level, or retrospective uncertainty calibration. The unifying statistical problem is that naïve reuse of the same data for both selection and inference generally invalidates pre-selection guarantees, producing anti-conservative confidence intervals, inflated type I error, or misleading uncertainty statements. Recent work shows that valid post-hoc inference can nevertheless be constructed in several distinct ways: by conditioning on the selection event, by deriving simultaneous guarantees over all admissible post-selection targets, by calibrating uncertainty through e-values and post-hoc $p$-values, by exploiting max-information and randomized selection, or by building orthogonal estimating equations that are insensitive to nuisance-selection error (Shen et al., 16 Apr 2025, Blanchard et al., 2017, Chugg et al., 1 Aug 2025, Gradu et al., 2022, Belloni et al., 2013).

1. General statistical formulation

Post-hoc valid inference arises whenever an inferential target is chosen after inspecting the data. In multiple testing, this means making guarantees for any subset $R$ selected after seeing the $p$-values; in regression, it means constructing confidence intervals for coefficients after covariate selection; in causal discovery, it means estimating causal effects after learning a graph from the same sample; in modern uncertainty quantification, it may mean fitting an uncertainty layer after a predictor has already been trained (Blanchard et al., 2017, Gradu et al., 2022, Bramlage et al., 1 Jun 2025).

A central formulation in large-scale testing is the requirement that, with probability at least $1-\alpha$, a bound on false discoveries or a lower bound on true discoveries holds simultaneously for every subset that might be chosen post hoc. In the joint-family-wise-error-rate framework, one seeks a function $V(R)$ such that

$$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$

Equivalently, $S(R)=|R|-V(R)$ is a simultaneous lower bound on the number of true discoveries in any user-chosen set $R$ (Blanchard et al., 2017). This formulation is explicitly “user-agnostic”: the choice of $R$ may be arbitrary and data-dependent.

In regression post-selection problems, an analogous principle appears as simultaneous coverage over a family of candidate models. For linear regression, valid post-selection inference for every possible random model selector $\hat M$ is equivalent to simultaneous coverage over all candidate models $R$0 in the model universe. This equivalence converts a selective problem into a simultaneous one and underlies assumption-lean post-selection regions that remain valid under arbitrary variable selection (Kuchibhotla et al., 2018).

A different but related formulation concerns post-hoc choice of the significance level itself. Classical tests require $R$1 to be fixed before analysis. Post-hoc $R$2 testing instead permits $R$3 to depend on the data and replaces ordinary size control with a risk-control condition based on expected size distortion. In that setting, a $R$4-value is post-hoc valid if and only if its reciprocal is an e-value (Koning, 2023), and recent work extends this logic to large-sample confidence sets and $R$5-values via asymptotic e-values and e-processes (Chugg et al., 9 Mar 2026, Chugg et al., 1 Aug 2025).

2. Principal methodological paradigms

Several non-equivalent methodologies fall under the heading of post-hoc valid inference. They differ mainly in what object is conditioned on, what quantity is guaranteed, and whether validity is finite-sample or asymptotic.

Selective inference via conditioning on the selection event treats selection as part of the stochastic experiment. In LASSO-type problems, the Karush–Kuhn–Tucker conditions can often express the event “model $R$6 was selected” as a polyhedron $R$7 in a Gaussian statistic $R$8. The polyhedral lemma then yields a truncated normal law for a linear functional $R$9, enabling conditional $p$0-values and confidence intervals for selected coefficients. In the group-testing logistic regression setting, this approach is extended to partially observed responses by combining EM-based asymptotic normality with a polyhedral approximation to the selection event (Shen et al., 16 Apr 2025).

Simultaneous inference over all candidate targets dispenses with explicit conditioning. In large-scale testing, the joint family-wise error rate defines a reference family of thresholded rejection sets and yields a simultaneous bound $p$1 for all subsets $p$2. In assumption-lean linear regression, deterministic inequalities for OLS normal equations produce model-wise confidence regions whose simultaneous validity over all models implies valid post-selection coverage for every data-driven model selector (Blanchard et al., 2017, Kuchibhotla et al., 2018).

Orthogonal-score or debiasing approaches make the inferential moment locally insensitive to nuisance estimation and model-selection mistakes. In high-dimensional approximately sparse quantile regression, the score for the low-dimensional parameter $p$3 is orthogonal to first-order perturbations in the nuisance functions $p$4 and $p$5. This orthogonality protects inference against moderate variable-selection errors and yields uniform validity without assuming perfect model recovery (Belloni et al., 2013).

Randomized selection plus information control constructs validity by limiting how much the selection stage can overfit the data. In inference after causal discovery, a differentially private randomized version of score-based graph search or GES is used to control approximate max-information between the selected graph and the sample. This produces a correction factor $p$6 such that naïve fixed-graph confidence intervals become valid after randomized graph discovery (Gradu et al., 2022).

Post-hoc uncertainty retrofitting addresses predictive uncertainty rather than hypothesis testing. In regression networks, a frozen point-estimating model $p$7 is augmented by an auxiliary model $p$8 trained post hoc using the detached Gaussian negative log-likelihood. Under the Gaussian heteroscedastic model, this is exactly the second stage of the canonical MLE for $p$9 given a fixed mean, so the resulting predictive variance estimator is statistically grounded despite the base model being treated as a black box (Bramlage et al., 1 Jun 2025).

Permutation- and conformal-based post-hoc procedures support arbitrary data-driven region selection or stopping decisions. In fMRI, ARI, Notip, and pARI provide valid lower bounds on the true discovery proportion in any cluster or subcluster selected after inspection of the statistical map (Peyrouset et al., 4 Nov 2025). In LLM reasoning, conformal thresholds calibrated on graph-level factuality scores can stop generation at inference time while maintaining coverage guarantees for “no-false” or “no-miss” factuality objectives (Wang et al., 7 Jun 2026).

3. E-values, post-hoc $1-\alpha$0-values, and data-dependent significance levels

A major recent development is the recognition that post-hoc valid hypothesis testing is naturally expressed in terms of e-values. The basic fixed-hypothesis characterization is that a $1-\alpha$1-value $1-\alpha$2 is post-hoc valid if and only if

$1-\alpha$3

under the null; equivalently, $1-\alpha$4 is an e-value (Koning, 2023). This converts the problem of allowing data-dependent $1-\alpha$5 into one of constructing nonnegative random variables with null expectation at most one.

The resulting rejection rule is not tied to any pre-specified threshold. If $1-\alpha$6 is an e-value, the rule “reject when $1-\alpha$7” is valid for every fixed $1-\alpha$8, and because $1-\alpha$9 is post-hoc valid, $V(R)$0 may itself be chosen after seeing the data without inducing size distortion in expectation (Koning, 2023). This directly addresses the “roving $V(R)$1” problem, in which analysts adjust the significance level after observing how close a result is to conventional cutoffs (Chugg et al., 1 Aug 2025, Chugg et al., 9 Mar 2026).

The post-hoc testing framework can be formalized using a family of losses indexed by $V(R)$2, where $V(R)$3 encodes the effective cost of type I versus type II errors. A test family $V(R)$4 is type-I risk safe if

$V(R)$5

Within this framework, $V(R)$6-admissibility asks whether a test can be uniformly improved over a class $V(R)$7 of adversaries that choose $V(R)$8 after seeing the data. For point null and point alternative hypotheses, every $V(R)$9-admissible post-hoc test must be determined by a sharp e-value, and for broad adversary classes this characterization is essentially complete (Chugg et al., 1 Aug 2025).

Large-sample analogues have now been developed. An asymptotic e-variable $$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$0 satisfies

$$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$1

and asymptotic post-hoc confidence intervals are exactly level sets of such asymptotic e-values under mild monotonicity conditions (Chugg et al., 9 Mar 2026). For mean inference, self-normalized constructions based on

$$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$2

yield asymptotic post-hoc confidence intervals whose width scales like $$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$3, matching Wald behavior in order while allowing post-hoc choice of $$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$4 (Chugg et al., 9 Mar 2026).

This suggests a general principle: whenever post-hoc validity refers to choosing the evidence threshold after looking at the data, e-values are the natural primitive, and post-hoc $$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$5-values are their reciprocals (Koning, 2023, Chugg et al., 1 Aug 2025, Chugg et al., 9 Mar 2026, Koning, 2023).

4. Post-selection inference in regression

Regression remains the canonical domain of post-hoc valid inference because model selection is routine and naïve inference after selection is usually anti-conservative. Several strands of work address different versions of this problem.

In assumption-lean linear regression, the target parameter for a model $$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$6 is the population OLS projection

$$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$7

with no assumption that the linear model is correctly specified. Valid post-selection inference is defined as coverage of $$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$8 for every random selector $$\mathbb P_{X\sim P}\bigg( \forall R\subset N,\; |R\cap \mathcal H_0(P)| \le V(R) \bigg) \;\ge\; 1-\alpha .$$9. The key theorem establishes that this is equivalent to simultaneous coverage of $S(R)=|R|-V(R)$0 for all $S(R)=|R|-V(R)$1 in the model collection. Deterministic inequalities involving global covariance and cross-moment deviations then yield confidence regions valid for all models simultaneously, and hence post-selection valid for arbitrary selection procedures (Kuchibhotla et al., 2018).

In high-dimensional approximately sparse quantile regression, the parameter of interest is a low-dimensional component $S(R)=|R|-V(R)$2 in a model of the form

$S(R)=|R|-V(R)$3

The nuisance $S(R)=|R|-V(R)$4 is approximated by a high-dimensional basis $S(R)=|R|-V(R)$5, and a density-weighted auxiliary regression

$S(R)=|R|-V(R)$6

produces an orthogonal score whose derivative with respect to the nuisance parameters vanishes at the truth. This orthogonality means that selection or estimation errors in $S(R)=|R|-V(R)$7 and $S(R)=|R|-V(R)$8 affect the target only in second order. Both the explicit orthogonal-score estimator and the weighted double-selection estimator achieve root-$S(R)=|R|-V(R)$9 asymptotic normality and uniform coverage over large model classes without requiring perfect support recovery (Belloni et al., 2013).

In logistic regression with group testing data, the challenge is more elaborate because the binary outcomes $R$0 are latent and only pooled, error-prone test outcomes $R$1 are observed. The method first fits an EM-based LASSO logistic regression, with pseudo-responses $R$2. Variable selection is represented via KKT conditions in the EM surrogate, and a pre-selection estimator $R$3 is shown to satisfy an approximate Gaussian law with a sandwich variance estimator. The polyhedral lemma is then applied to the approximate Gaussian statistic, yielding truncated-normal post-selection $R$4-values and confidence intervals for selected coefficients (Shen et al., 16 Apr 2025). Simulation evidence indicates that naïve post-selection Wald intervals can have empirical coverage around $R$5–$R$6 instead of $R$7, whereas the selective procedure restores coverage near $R$8, albeit with wider intervals (Shen et al., 16 Apr 2025).

These regression examples illustrate two broad routes to validity: either represent selection explicitly and condition on it, or construct orthogonal or simultaneous inferential objects that remain stable under imperfect selection (Shen et al., 16 Apr 2025, Kuchibhotla et al., 2018, Belloni et al., 2013).

5. Post-hoc inference in multiple testing, imaging, and exploratory set selection

Post-hoc valid inference is particularly consequential in settings where analysts inspect complex output structures and then choose interesting subsets for formal claims. Functional neuroimaging is a leading example.

All Resolutions Inference (ARI) provides simultaneous lower bounds on the true discovery proportion (TDP) for every voxel subset $R$9. If $R$0 denotes the true null voxels, then

$R$1

and ARI delivers a lower confidence bound $R$2 such that

$R$3

Because the quantifier “for all $R$4” lies inside the probability statement, the analyst may choose $R$5 after seeing the data—large cluster, subcluster, or arbitrary region of interest—without invalidating the claim (Peyrouset et al., 4 Nov 2025).

The JER framework provides the abstract multiple-testing machinery behind these guarantees. For a threshold family $R$6, the Joint Error Rate is

$R$7

and JER control implies valid post-hoc TDP bounds through an interpolation formula (Peyrouset et al., 4 Nov 2025, Blanchard et al., 2017). This framework is explicitly “user-agnostic”: once the threshold family is calibrated, any set $R$8 may be selected and reported (Blanchard et al., 2017).

Permutation-calibrated methods such as Notip and pARI sharpen ARI by adapting to the strong dependence structure of fMRI data. Their comparison reveals that power depends strongly on cluster size. With $R$9, pARI achieves higher sensitivity on large clusters, but Notip provides more informative and robust bounds on smaller clusters and is better suited to “drill-down” exploration into subregions at high cluster-forming thresholds (Peyrouset et al., 4 Nov 2025). This is a concrete instance of a general tradeoff in post-hoc valid inference: the form of simultaneous guarantee may be fixed, but the practical informativeness of the resulting bounds depends on how power is allocated across the space of possible post-hoc queries.

This suggests a broader interpretation. Post-hoc valid inference in exploratory science often amounts to replacing single-decision error control with a simultaneous envelope over all decisions the analyst might plausibly make after inspection (Blanchard et al., 2017, Peyrouset et al., 4 Nov 2025).

6. Causal discovery, missing microdata, and other nonstandard settings

Recent work extends post-hoc validity beyond classical regression and multiple testing to settings where the inferential target itself is generated by a complex preprocessing or discovery step.

After causal discovery, naïve confidence intervals for causal effects are invalid because the selected graph $\hat M$0 depends on the same sample used for downstream estimation. To address this, randomized versions of exhaustive score-based search and greedy equivalence search are made differentially private, which bounds the approximate max-information $\hat M$1 between the selected graph and the dataset. If $\hat M$2 is chosen as

$\hat M$3

for the randomized exhaustive search case, then naïve fixed-graph confidence intervals evaluated at level $\hat M$4 become valid after graph discovery (Gradu et al., 2022). This route to post-hoc validity does not require explicit conditioning on the graph-selection event and is compatible with complex search procedures.

When only published point estimates and standard errors are available, valid inference on nonlinear functionals of causal effects is obstructed by missing covariances. The proposed solution is to maximize the delta-method variance over all positive semidefinite covariance matrices consistent with the reported marginal standard errors. The worst-case asymptotic variance can be formulated as a semidefinite program and, in the unconstrained case, has the analytic form

$\hat M$5

Using $\hat M$6 in place of the unknown true standard deviation yields conservative but valid confidence intervals for $\hat M$7, including for policy metrics such as the Marginal Value of Public Funds (Vohra, 2024). This is post-hoc valid inference in the sense that valid conclusions can be drawn from limited summary information after the original study is complete and the microdata are inaccessible.

In post-hoc dataset inference for LLMs, the inferential question is whether a suspect dataset was used in training. Because no real held-out in-distribution set is available post hoc, a synthetic held-out set is generated via a suffix-completion generator, and then a dual-classifier calibration removes the distribution shift between natural and synthetic data. The final test compares the extra discriminative power conferred by membership-inference signals beyond what can be explained by text-generation artifacts alone (Zhao et al., 18 Jun 2025). This case differs from classical statistical selection problems, but it fits the same pattern: a nuisance post-hoc construction step introduces bias, and a second calibration layer is needed to restore type I reliability.

These examples indicate that post-hoc validity is not tied to a single inferential object. It can target regression coefficients, causal functionals, graph-conditional effects, or dataset-membership hypotheses, provided one can either characterize the post-hoc dependence or dominate it by a worst-case or randomized argument (Gradu et al., 2022, Vohra, 2024, Zhao et al., 18 Jun 2025).

7. Post-hoc predictive uncertainty and structured reasoning

Another contemporary branch of post-hoc valid inference concerns predictive rather than parameter uncertainty. Here the inferential question is not whether a coefficient differs from zero, but whether a prediction set, variance estimate, or factuality score is reliable after being added retrospectively to a model.

In regression networks, the IO-CUE framework trains a frozen predictor $\hat M$8 as a point estimator and then fits an auxiliary variance model $\hat M$9 by minimizing the detached Gaussian negative log-likelihood

$R$00

Because the mean is held fixed, this is exactly the conditional Gaussian MLE for $R$01 under the heteroscedastic model. The method therefore recovers input-dependent aleatoric uncertainty in a post-hoc manner while also extracting a quasi-epistemic signal from the geometry of the frozen outputs $R$02 (Bramlage et al., 1 Jun 2025). The notion of validity here is calibration of the predictive distribution rather than coverage after selection, but the conceptual theme is similar: a second-stage inferential layer is added after the original model has been trained, and its objective must be chosen so that statistical meaning is preserved.

For multi-step LLM reasoning, conformal methods have recently been adapted from post-hoc pruning to inference-time stopping. Reasoning traces are represented as DAGs, and a graph-level factuality uncertainty score $R$03 is converted into a nonconformity score

$R$04

If $R$05 is chosen so that $R$06 is nested along the sequence of expanding subgraphs, conformal quantiles calibrated on “earliest bad subgraphs” or “minimal full-true subgraphs” yield finite-sample coverage guarantees for no-false and no-miss factuality objectives (Wang et al., 7 Jun 2026). This can be viewed as a structured-output extension of post-hoc conformal prediction in which the “post-hoc” object is not a selected coefficient but a generated reasoning graph.

A plausible implication is that post-hoc valid inference increasingly includes procedures that retrofit formal uncertainty semantics onto already-trained black-box systems. In these settings, the key requirement is not necessarily conditioning on a selection event, but designing the second-stage inferential criterion so that its optimization or calibration corresponds to a proper statistical target (Bramlage et al., 1 Jun 2025, Wang et al., 7 Jun 2026).

8. Validity guarantees, efficiency tradeoffs, and recurring limitations

Across these literatures, “validity” is not uniform in meaning. It may denote exact conditional coverage after selection, simultaneous unconditional coverage over all possible post-hoc choices, bounded expected size distortion under data-dependent $R$07, or calibrated predictive uncertainty. What these notions share is robustness to a specified class of data-dependent analyst actions.

Finite-sample exactness is rare outside special Gaussian or permutation settings. The polyhedral-lemma framework yields exact conditional inference when the selected statistic is Gaussian and the selection event is exactly polyhedral, but in latent-variable group-testing logistic regression the method is only asymptotically valid because Gaussianity and polyhedrality enter through approximations (Shen et al., 16 Apr 2025). ARI-style set-wise bounds and JER-based procedures are finite-sample valid under their dependence assumptions, and conformal reasoning also retains finite-sample coverage under exchangeability (Peyrouset et al., 4 Nov 2025, Wang et al., 7 Jun 2026). Asymptotic e-value methods, orthogonal-score methods, and summary-statistics worst-case delta methods are asymptotic rather than exact (Chugg et al., 9 Mar 2026, Belloni et al., 2013, Vohra, 2024).

Efficiency loss is the main price of validity. Selective intervals are wider than naïve ones because they condition on the selection event (Shen et al., 16 Apr 2025). Simultaneous regions over all models or all subsets are wider than procedure-specific or target-specific intervals (Blanchard et al., 2017, Kuchibhotla et al., 2018). Data splitting is broadly valid but typically much less efficient than randomized max-information corrections or orthogonal-score procedures (Gradu et al., 2022, Shen et al., 16 Apr 2025). Post-hoc e-value methods can approximate classical asymptotic widths in large samples, but they remain somewhat more conservative than fixed-$R$08 Wald procedures because they solve a stronger problem (Chugg et al., 9 Mar 2026, Chugg et al., 1 Aug 2025, Koning, 2023).

Several recurring limitations appear across domains.

Model dependence remains central. Group-testing post-selection inference depends on correct specification of the logistic model and test sensitivity/specificity (Shen et al., 16 Apr 2025). Summary-statistics inference depends on the reliability of reported standard errors and the differentiability of the target functional (Vohra, 2024). IO-CUE assumes Gaussian noise and a good mean estimator (Bramlage et al., 1 Jun 2025).

Computation can be substantial. Simultaneous model-wise post-selection regions in regression rely on bootstrap approximations in $R$09-dimensional moment spaces (Kuchibhotla et al., 2018). Randomized causal discovery requires repeated scoring of candidate graphs under noisy mechanisms (Gradu et al., 2022). Selective EM-based post-selection inference combines penalized optimization, variance estimation, and truncated-normal inversion (Shen et al., 16 Apr 2025).

Scope of validity must be interpreted carefully. Post-hoc valid testing with e-values controls expected size distortion, not ordinary conditional size at every realized $R$10 (Koning, 2023, Chugg et al., 1 Aug 2025). Worst-case covariance methods yield conservative intervals for the actual target only if the unknown covariance belongs to the feasible semidefinite set (Vohra, 2024). Predictive calibration does not imply Bayesian epistemic uncertainty unless the corresponding structural assumptions hold (Bramlage et al., 1 Jun 2025).

A common misconception is that “valid post-hoc inference” means one can perform arbitrary data mining with no inferential cost. The literature shows the opposite. Validity is achievable only when the inferential procedure is expanded to account for the post-hoc action—through conditioning, simultaneity, orthogonality, calibration, or randomization—and that expansion almost always incurs some loss of sharpness or computational simplicity (Blanchard et al., 2017, Kuchibhotla et al., 2018, Shen et al., 16 Apr 2025, Gradu et al., 2022).

9. Conceptual synthesis and current direction

A coherent way to organize the field is to distinguish what is being protected against.

When the post-hoc action is selection of a model or subset, validity is often achieved by simultaneous coverage or selective conditioning (Blanchard et al., 2017, Kuchibhotla et al., 2018, Shen et al., 16 Apr 2025, Peyrouset et al., 4 Nov 2025). When it is estimation of nuisance functions after variable selection, orthogonality becomes the key robustness device (Belloni et al., 2013). When it is choosing the significance level after seeing the data, e-values and post-hoc $R$11-values provide the correct inferential currency (Koning, 2023, Chugg et al., 1 Aug 2025, Chugg et al., 9 Mar 2026). When it is retroactively adding uncertainty to a predictor or generator, proper likelihood or conformal calibration is essential (Bramlage et al., 1 Jun 2025, Wang et al., 7 Jun 2026). When it is using the same data for discovery and estimation, randomized selection plus information control offers another route (Gradu et al., 2022).

This suggests an underlying principle. Post-hoc valid inference requires moving the inferential object one level higher than in standard analyses. Instead of reasoning about a single model, coefficient, threshold, or prediction rule, one reasons about the entire family of data-dependent actions that may be taken after inspection. The guarantee must hold uniformly, conditionally, or adaptively with respect to that enlarged action space. The relevant mathematics then depends on the structure of the action space: polyhedra for LASSO events, simultaneous envelopes for arbitrary subsets, semidefinite sets for unknown covariance matrices, martingales for anytime validity, or orbit averages for invariance testing (Shen et al., 16 Apr 2025, Blanchard et al., 2017, Vohra, 2024, Chugg et al., 9 Mar 2026, Koning, 2023).

Current research is therefore broadening rather than narrowing the concept. Post-hoc validity now encompasses classical selective inference, adaptive error-control via e-values, inference after causal discovery, robust inference from summary statistics, post-hoc predictive uncertainty for neural networks, and inference-time calibration for structured generative models (Chugg et al., 1 Aug 2025, Gradu et al., 2022, Vohra, 2024, Bramlage et al., 1 Jun 2025, Wang et al., 7 Jun 2026). The common theme is that data dependence is not ignored but elevated into the design criterion of the inferential procedure itself.