Post-model-selection predictor PoSI confidence intervals under full and active-only covariate observation
Mathematical Methods AIMech:PostModelSelectionPredictorPoSIConfidenceIntervals DRAFT
Aliases: Bachoc-Leeb-Pötscher post-model-selection predictor confidence intervals, Active-covariate partial-union-bound predictor PoSI hierarchy (K_1, K_2, K_3, K_4, K_5), Design-dependent in-sample and random-design out-of-sample post-selection prediction intervals, Post-selection linear predictor simultaneous coverage calibration
A post-model-selection inference methodology for linear regression predictors x_0'[M_hat] beta_hat_{M_hat} +- K(x_0, M_hat) * ||s_{M_hat}|| * sigma_hat (Equation 5), where s_M' = x_0'[M] (X[M]' X[M])^{-1} X[M]' (Equation 6) and bar{s}_M = s_M / ||s_M||, guaranteeing coverage >= 1 - alpha simultaneously over all candidate models M in M_cal and any data-driven model selector M_hat without requiring the true mean mu to be linear. In fixed-design homoscedastic Gaussian regression Y ~ N(mu, sigma^2 I_n) targeting the in-sample optimal linear predictor x_0'[M_hat] beta_{M_hat}^{(n)} with beta_M^{(n)} = (X[M]' X[M])^{-1} X[M]' mu (Equations 3-4, Remark 2.8), observing the full regressor vector x_0 in R^p yields the exact full-information constant K_1(x_0, r), the (1 - alpha)-quantile of max_{M in M_cal} |bar{s}_M' (Y - mu)| / sigma_hat =^d G * max_{M in M_cal} |bar{s}_M' V| (Equation 10, Proposition 2.1, Equation 23). When only the selected covariates x_0[M_hat] are observed at prediction time, K_1(x_0) is infeasible and is replaced by the active-covariate hierarchy K_naive <= K_1(x_0) <= K_2(x_0[M], M) <= K_3(x_0[M], M) <= K_4 <= K_5 (Equations 11-15, Proposition 2.3, Corollary 2.4), where K_2(x_0[M], M) = sup_{x[M] = x_0[M]} K_1(x), K_3(x_0[M], M) is the (1 - alpha)-quantile of the partial spherical-cap union-bound distribution F_{M, x_0}(t) = E_G[1 - min(1, P(max_{M_* subset M} |bar{s}_{M_*}' V| > t/G) + c(M, M_cal) * (1 - F_{Beta, 1/2, (d-1)/2}((t/G)^2)))] separating nested submodels M_* subset M from the c(M, M_cal) non-nested models (Equations 12-14), K_4 = K_3(x_0[emptyset], emptyset) is the full union-bound constant, and K_5 is the Scheffé constant. In random-design regression Y = X beta + U with second-moment matrix Sigma, the same intervals (with either full-model or post-selection variance estimator sigma_hat_{M_hat}^2) asymptotically uniformly cover the design-independent out-of-sample optimal linear predictor x_0'[M_hat] beta_{M_hat}^{(*)} with beta_M^{(*)} = beta[M] + (Sigma[M, M])^{-1} Sigma[M, M^c] beta[M^c] (Section 3, Remark 3.2, Theorem 3.6, Lemma C.2, Theorem D.6).
Identity Notes & Concept Scope
Distinct from AIMech:ValidPostSelectionInferencePoSIConstant (Berk, Brown, Buja, Zhang, & Zhao 2013, The Annals of Statistics 41(2):802-837, arXiv:1206.6451), which constructs simultaneous confidence intervals for individual submodel regression coefficients beta_{j, M} (corresponding to the sparse unit-vector special case x_0 = e_j) and shows that the coefficient PoSI constant scales as O(sqrt(log p)) under orthogonal designs. Bachoc, Leeb, & Pötscher (2019, The Annals of Statistics 47(3):1475-1504, arXiv:1412.4605v3, DOI:10.1214/18-AOS1721) formulate post-selection inference for general linear predictors x_0'[M_hat] beta_{M_hat} under both full observation of x_0 (K_1(x_0)) and active-only observation of x_0[M_hat] (K_2, K_3, K_4), prove exact tightness of the partial union bound in Lemma B.4, establish that even under orthogonal designs the predictor constants K_1(x_0, infty) and K_2(x_0[M], M, infty) grow at the polynomial rate Theta(sqrt(p)) with explicit lower-bound constant xi = sup_{b > 0} phi(b)/sqrt(1 - Phi(b)) approx 0.6363 (Proposition 2.9) while K_3(x_0[M], M, infty) / K_4(infty) -> 1 and K_4(infty)/sqrt(p) -> sqrt(3)/2 approx 0.866 < 1 (Proposition 2.10, Corollary 2.12), and prove uniform asymptotic validity for the random-design out-of-sample projection target x_0'[M_hat] beta_{M_hat}^{(*)} (Theorem 3.6, Theorem D.6). Also distinct from AIMech:RestrictedIsometryPostSelectionInferenceConstantBounds (Bachoc, Blanchard, & Neuvial 2018), which bounds the regression-coefficient PoSI constant under s-sparse Restricted Isometry Property assumptions.
Claims & Grounded Evidence
4 Claims
Mechanism
#full-and-active-covariate-predictor-posi-constant-hierarchy
For fixed-design homoscedastic Gaussian regression Y ~ N(mu, sigma^2 I_n) with candidate model set M_cal subset 2^{{1, ..., p}}, rank d = rank(X), and variance estimator sigma_hat^2 independent of X'Y distributed as (sigma^2 / r) chi^2_r (including r = infty when sigma is known), the post-model-selection predictor confidence interval CI(x_0) = x_0'[M_hat] beta_hat_{M_hat} +- K(x_0, M_hat) * ||s_{M_hat}|| * sigma_hat (Equations 5-6) covers the design-dependent in-sample optimal target x_0'[M_hat] beta_{M_hat}^{(n)} with probability >= 1 - alpha for every model selector M_hat whenever K(x_0, M_hat) >= K_1(x_0), where K_1(x_0) is the (1 - alpha)-quantile of max_{M in M_cal} |bar{s}_M'(Y - mu)| / sigma_hat =^d G * max_{M in M_cal} |bar{s}_M' V| with G ~ sqrt(d/r) * sqrt(chi^2_d / chi^2_r) independent of V ~ Unif(S^{d-1}) (Equation 10, Proposition 2.1, Corollary 2.2, Equation 23). When only active covariates x_0[M_hat] are observed, Proposition 2.3 and Corollary 2.4 (pages 9-10, Equations 11-17) establish the valid constant hierarchy K_naive = q_{r, 1-alpha/2} <= K_1(x_0) <= K_2(x_0[M], M) <= K_3(x_0[M], M) <= K_4 <= K_5 and the anti-monotonicity K_i(x_0[M_2], M_2) <= K_i(x_0[M_1], M_1) for M_1 subset M_2 (i = 2, 3), where K_2(x_0[M], M) = sup{K_1(x) : x[M] = x_0[M]} (Equation 11) and K_3(x_0[M], M) is the (1 - alpha)-quantile of F_{M, x_0}(t) = E_G[F^*_{M, x_0}(t/G)] with F^*_{M, x_0}(t) = 1 - min(1, P(max_{M_* in M_cal, M_* subset M} |bar{s}_{M_*}' V| > t) + c(M, M_cal) * (1 - F_{Beta, 1/2, (d-1)/2}(t^2))) (Equations 12-14).
Mechanism
#exact-union-bound-geometry-and-monte-carlo-quantile-algorithms
Lemma B.4 (Appendix B.1, page 34) proves that the spherical union bounds defining K_3(x_0[M], M) and K_4 can hold with exact equality: for p = d = 2, n >= 2, M_cal = 2^{{1, 2}}, X^{(2)} = [[1, cos(2*pi/3)], [0, sin(2*pi/3)]], and x_0' = (cos(4*pi/3), sin(4*pi/3)) X^{(2)}, the three unit vectors bar{s}_{{1}}' = -(1, 0), bar{s}_{{2}}' = -(cos(2*pi/3), sin(2*pi/3)), and bar{s}_{{1,2}}' = (cos(4*pi/3), sin(4*pi/3)) have spherical caps {|bar{s}_{M_*}' V| > c} that are pairwise disjoint on S^1 for c >= sqrt(3)/2 and cover S^1 for 0 <= c <= sqrt(3)/2, yielding F^*_{M, x_0}(t) = P(max_{M_* in M_cal} |bar{s}_{M_*}' V| <= t) and K_1(x_0) = K_2(x_0[M], M) = K_3(x_0[M], M) = K_4 < K_5 for every M in M_cal. Appendix E (pages 48-53, Equations 37-42, Algorithms E.1-E.3) provides exact 1D numerical integration for K_4 (Equation 37) and conditional Monte Carlo integration over V ~ Unif(S^{d-1}) and V_tilde ~ Unif(S^{|M|-1}) for K_1(x_0) and K_3(x_0[M], M), reducing the inner nested maximization in K_3 from dimension d to dimension |M| via QR orthogonalization of X[M] (Equation 40, Algorithm E.2).
Mechanism
#orthogonal-design-sqrt-p-scaling-and-universal-k3-k4-asymptotics
Whereas the Berk-Brown-Buja-Zhang-Zhao (2013) PoSI constant for individual regression coefficients scales as sqrt(2 log p) under orthogonal designs, Proposition 2.9 and Lemma B.6 (Section 2.4.1, pages 12-13, Appendix B.2, pages 35-36) prove that for any orthogonal design X(p) in X_sf(p) with M_cal = 2^{{1, ..., p}} and r = infty, choosing dense x_0 proportional to (1, ..., 1)' yields liminf_{p -> infty} K_1(x_0, infty) / sqrt(p) >= xi = sup_{b > 0} phi(b) / sqrt(1 - Phi(b)) approx 0.6363 (while sparse x_0 with sup_p ||x_0||_0 < infty satisfies K_1(x_0, infty) = O(1)), and for any 0 <= gamma < 1, liminf_{p -> infty} inf_{x_0, X in X_sf(p), |M| <= gamma * p} K_2(x_0[M], M, infty) / sqrt(p) >= xi * sqrt(1 - gamma). Furthermore, Proposition 2.10 and Corollaries 2.11-2.12 (Section 2.4.2, pages 13-14, Appendix B.2, pages 36-38) prove that whenever c(M, M_cal) >= tau * |M_cal| (holding with tau = 1/3 for all M != {1, ..., p} when M_cal = {M : |M| <= m_p}), K_3(x_0[M], M, r_p) / K_4(r_p) -> 1 uniformly over design X, x_0, and M, and K_4(infty) ~ sqrt(min(n(p), p) * (1 - |M_cal|^{-2/(min(n(p), p) - 1)})), which for M_cal = 2^{{1, ..., p}} and n(p) >= p gives lim_{p -> infty} K_4(infty) / sqrt(p) = sqrt(3)/2 approx 0.866 < 1 = lim_{p -> infty} K_5(infty) / sqrt(p).
Mechanism
#random-design-uniform-asymptotic-validity-for-out-of-sample-predictors
In the random-design regression model Y = X beta + U where (y_i, x_i') are i.i.d. with E[x_i x_i'] = Sigma and E[u_i | x_i] = 0, Remark 2.8 and Section 3 (pages 11-12, 15-19, Equation 18, Remark 3.2) distinguish the design-dependent in-sample projection parameter beta_M^{(n)} = (X[M]' X[M])^{-1} X[M]' E[Y | X] from the design-independent out-of-sample optimal linear projection parameter beta_M^{(*)} = argmin_{gamma} E[(y_0 - x_0'[M] gamma)^2] = beta[M] + (Sigma[M, M])^{-1} Sigma[M, M^c] beta[M^c]. Under Conditions 3.3-3.5 (moment bounds on X, U and spherical invariance of X Sigma^{-1/2} when |M_hat| < p), Theorem 3.6 and Lemmas C.1-C.2 (Section 3, pages 17-19, Appendix C, pages 38-42) prove that the confidence intervals CI(x_0) calibrated with K(x_0, M_hat) in {K_1, K_2, K_3, K_4, K_5} and either the full-model variance estimator sigma_hat_OLS^2 or the post-selection variance estimator sigma_hat_{M_hat}^2 = ||Y - X[M_hat] beta_hat_{M_hat}||^2 / (n - |M_hat|) achieve uniform asymptotic coverage P(x_0'[M_hat] beta_{M_hat}^{(*)} in CI(x_0) | x_0) >= (1 - alpha) + o_p(1) uniformly over model selectors M_hat and spherical error-plus-design laws (extended to growing dimension p(n) -> infty with p(n) = o(n^{1/2}) in Appendix D, Theorem D.6, pages 42-48).
⚠️ Curated Limitations & Negative Results
- Unlike the Berk-Brown-Buja-Zhang-Zhao (2013) PoSI constant for individual regression coefficients, which scales as sqrt(2 log p) under orthogonal designs, the predictor PoSI constants K_1(x_0) (for dense x_0) and K_2(x_0[M], M), K_3(x_0[M], M), K_4 grow at rate Theta(sqrt(p)) even under orthogonal designs when M_cal = 2^{{1, ..., p}}, because the predictor vectors bar{s}_M depend continuously on x_0 across all 2^p - 1 submodels (Proposition 2.9, Proposition 2.10, Corollary 2.12).
- In the random-design setting (Section 3, Condition 3.5, Remark 3.7, and Appendix D Condition D.4), uniform asymptotic validity over misspecified submodels |M_hat| < p requires spherical invariance of the standardized covariate vector Sigma^{-1/2} x_i (such as elliptical / Gaussian designs) to ensure that the omitted-variable projection residual x_i'[M^c] beta[M^c] - x_i'[M] (Sigma[M, M])^{-1} Sigma[M, M^c] beta[M^c] is uncorrelated with x_i[M] and conditionally homoscedastic.
- Four minor typographical or corrigendum notes in arXiv:1412.4605v3 were identified and verified during full-text inspection: (1) Section 2.4.1 (page 13, line 6) explicitly notes a corrigendum to Berk et al. (2013a) Proposition 5.5, pointing out that the remainder term printed as 'o(d)' in Berk et al. (2013a) should read 'o(1)'; (2) in Appendix B.2 (page 37, proof of Proposition 2.10, line 6), 'K(|M| - 1, min(n(p).p), alpha)' prints a period instead of a comma inside 'min(n(p), p)'; (3) in Appendix E (page 49, paragraph preceding Algorithm E.2, line 4), '\tilde{\bar{s}}_Minstead of \bar{s}_M' omits the space before 'instead'; and (4) in the References (page 56, Tian & Taylor 2015), 'selective inferene' is printed for 'selective inference'.
- All 56 pages of the full-text PDF of Bachoc, Leeb, & Pötscher (2019, The Annals of Statistics 47(3):1475-1504, arXiv:1412.4605v3, DOI:10.1214/18-AOS1721) were read. Constructed numerical checks in research/discovery/post-model-selection-predictor-posi-confidence-intervals-boundaries.py verify (1) Lemma B.4 exact union-bound geometry on S^1 where K_1(x_0) = K_2 = K_3 = K_4 < K_5, (2) the strict hierarchy and anti-monotonicity K_naive < K_1 < K_3(M_2) < K_3(M_1) = K_4 < K_5 for |M_1| = 1 < |M_2| = 2, (3) Proposition 2.9's orthogonal-design constant xi approx 0.636328 and Corollary 2.12's limit K_4(infty)/sqrt(p) -> sqrt(3)/2 approx 0.866025, and (4) Section 3's random-design coverage of the out-of-sample projection target beta_M^{(*)} using the post-selection variance estimator sigma_hat_{M_hat}^2.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:LinearRegression
Linear Regression
|
related_match | Constructs universally valid post-model-selection confidence intervals for linear regression predictors x_0'[M_hat] beta_hat_{M_hat} under both fixed and random designs (AIO:LinearRegression). |
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | Calibrates predictor confidence intervals via the PoSI hierarchy (K_1, K_2, K_3, K_4, K_5) to guarantee coverage >= 1 - alpha across all data-driven model selection procedures, preventing post-selection data-snooping invalidation (AIO:DataDredgingBias). |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
blp2019-pdf
|
François Bachoc, Hannes Leeb, Benedikt M. Pötscher
|
primary paper
full_text
|
ID: https://doi.org/10.1214/18-AOS1721
sha256:222cfb10a74f…
|
blp2019-crossref
|
François Bachoc, Hannes Leeb, Benedikt M. Pötscher
|
primary paper
abstract_only
|
ID: https://doi.org/10.1214/18-AOS1721
sha256:1b41a05db014…
|
blp2019-abs
|
François Bachoc, Hannes Leeb, Benedikt M. Pötscher
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.1412.4605
sha256:5df943653a28…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T17:35:00Z
Updated: 2026-10-17T17:35:00Z
Source File:
data/curated/PostModelSelectionPredictorPoSIConfidenceIntervals.yaml
· Schema:
curated_concept.yaml
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