Valid Bayesian Inference based on Variance Weighted Projection for High-Dimensional Logistic Regression with Binary Covariates

Abhishek Ojha; Naveen N. Narisetty

REVIEW 4 major objections 4 minor 15 references

Valid Bayesian Inference based on Variance Weighted Projection for High-Dimensional Logistic Regression with Binary Covariates

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that a variance-weighted projection score yields Bayesian credible intervals with correct frequentist coverage for a binary treatment effect in high-dimensional logistic regression.

desk verdict A genuinely new variance-weighted projection for binary treatments, but the coverage theorem rests on an assumption that quietly requires a correct propensity model. read the letter →

arxiv 2411.17618 v1 pith:MI3OO573 submitted 2024-11-26 stat.ME math.STstat.TH

Abhishek Ojha , Naveen N. Narisetty This is my paper · ORCID

classification stat.MEmath.STstat.TH MSC 62F1562J1262F12

keywords high-dimensionallogisticregressionBayesianinferenceNeymanorthogonalityvariance-weightedprojectioncredibleintervalsbinarytreatmentpropensityscoreconditionalposterior

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

High-dimensional logistic regression with a binary treatment and many confounders makes inference on the log-odds ratio hard because the usual orthogonality trick of regressing the treatment on the confounders breaks down. The paper constructs an orthogonal score that replaces the binary treatment with its variance-weighted projection onto the confounders, then builds a Bayesian conditional posterior for the treatment effect around that score. The main theorem states that this posterior is asymptotically normal at the oracle $n^{-1}$/2 rate, so the resulting credible intervals have correct frequentist coverage even when the number of confounders grows faster than the sample size. If true, this gives practitioners a Bayesian interval for treatment effects that avoids the bias of selection-based methods and does not require nonzero nuisance signals to be strong.

What carries the argument

The machinery is the variance-weighted projection h0(Z_i) = E[X_i Var(Y_i | X_i, Z_i) | Z_i] / E[Var(Y_i | X_i, Z_i) | Z_i], estimated from posterior samples of the nuisance parameters and a logistic working model for the propensity score. Subtracting this projection in the score (X - h0(Z))^T (Y - μ) forces the expected derivative with respect to the nuisance parameter β to vanish, giving Neyman orthogonality. The conditional posterior for θ then uses the working model with linear predictor (X_i - h(Z_i))θ + φ_i, which coincides exactly with the original logistic model at the oracle values; this is what lets the slower nuisance concentration rate be converted into the optimal $n^{-1}$/2 posterior rate for θ.

What would settle it

Run a simulation in which Y follows the assumed logistic model but the true probability that X equals 1 given Z is a nonlinear function, such as a probit or a logistic with interaction terms, that the linear logistic working model cannot represent; then compute the empirical coverage of the nominal 95% credible intervals over many replications. If coverage drops below nominal as n grows, the uniform propensity-concentration assumption fails and the theorem's conclusion is false.

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Extended reading notes

Core claim

The paper's central claim is that valid frequentist inference for the log-odds ratio of a binary covariate in a high-dimensional logistic model can be obtained from a Bayesian posterior built on an orthogonal score. Because the standard weighted regression of the binary treatment X on the nuisance covariates Z fails—the required moment condition would force a linear function to equal a bounded nonlinear ratio—the paper instead defines h0(Z_i) as the conditional expectation of X_i weighted by outcome variances, and uses the score (X - h0(Z))^T (Y - μ). Plugging posterior samples of the nuisance parameters into h and a reparametrized φ yields a conditional posterior for the treatment effect θ. Theorem 1 shows this marginal posterior converges in probability to a normal distribution centered at the oracle MLE θ̂0 with scale σ_n, and Corollary 1.1 concludes that the credible intervals have correct frequentist coverage.

Load-bearing premise

The load-bearing premise is that the logistic working model for the probability that the binary treatment equals 1 given the covariates estimates the true probability well enough that its error vanishes uniformly as the sample size grows, even though the paper does not assume that working model is correct; if this premise fails, the projection that removes nuisance bias is itself biased and the credible intervals can lose coverage.

Editorial extensions

If this is right

Credible intervals from the conditional posterior achieve nominal frequentist coverage for θ at the oracle n^-1/2 rate.
The method remains valid when the number of covariates grows sub-exponentially in n and does not require nonzero nuisance coefficients to be bounded away from zero.
The construction extends to a K-category treatment by applying the variance-weighted projection separately to each dummy variable.
Averaging over posterior samples of the nuisance parameters captures multiple orthogonality instances, in contrast to methods that rely on a single debiased estimate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same conditional-posterior construction should transfer to other high-dimensional generalized linear models with discrete exposures, with h0 defined by the same variance-weighted conditional expectation.
Because h0 is defined through conditional expectations, a nonparametric or doubly robust estimator of h0 could replace the logistic propensity working model and relax the paper's concentration assumption on the propensity score.
A practical user should treat the logistic propensity working model as a genuine assumption: in real datasets where the treatment depends nonlinearly on covariates, coverage should be checked empirically rather than taken for granted.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Punchline: the variance-weighted projection in (2.9) is a real idea, and the paper is right that the continuous-X orthogonality trick fails when X is binary. The trouble is that Theorem 1 and Corollary 1.1 lean on Assumption A5, which is closer to assuming a correct propensity model than the paper admits. That's the main soft spot, and it's load-bearing.

What's new: h0(Z_i) = E[X_i Var(Y_i|X_i,Z_i)|Z_i] / E[Var(Y_i|X_i,Z_i)|Z_i] is a novel construction, and the derivation from the conditional moment equation (8.2) is clean. Combining this with the conditional-posterior framework is a reasonable step forward. The simulations are reasonably thorough; CB's coverage is close to nominal across the grid and competitive with the oracle and LSW. That's real evidence the method works when the propensity model is correctly specified.

Soft spots, in rough order of severity. First, A5 requires the estimated log odds of the propensity to be uniformly within M sqrt(log(d∨n)/n) of the truth. If the logistic working model for X|Z is misspecified, the posterior for γ concentrates at the pseudo-true value, and the log-odds bias is generically of constant order for at least some individuals, so A5 fails. The paper says it doesn't assume a correct model, but then imposes exactly the kind of parametric-rate closeness that a correct model would deliver. Section 5.3 concedes the working model 'might not strictly apply,' and Section 6.2 uses a LASSO shortcut not covered by Assumption A4, so the empirical sections don't rescue the abstract's unqualified 'valid inference' claim. Second, the definition of sigma_n in (3.9) has the opposite sign from the proof in (B.16). That's fixable, but as written the proof is internally inconsistent. Third, Lemma 7's proof applies a weak law to terms that are dependent through the estimated nuisance functions; a rigorous treatment needs to condition on a high-probability set more explicitly. The overall strategy of proving the theorem on a high-probability set is standard and fine.

For whom: this is for a reader working on Bayesian inference for low-dimensional parameters in high-dimensional GLMs, especially with binary/categorical treatments. The projection idea deserves attention; the theorem as stated doesn't fully support the coverage claim. I'd send this to peer review, with a strong request to fix A5 (either prove it under explicit conditions or weaken the claims), correct the sigma_n and dependence issues, and provide code and data. With those fixes it could be a solid contribution.

Referee Report

4 major / 4 minor

Summary. The paper proposes a new Bayesian method for inference on a low-dimensional parameter θ in a high-dimensional logistic regression model where the covariate of interest X is binary. The central construction is an orthogonal score based on a variance-weighted projection h0(Z) = E[X Var(Y|X,Z) | Z] / E[Var(Y|X,Z) | Z], which the authors argue is needed because the usual weighted-regression orthogonality condition fails for binary X. The method draws nuisance parameters (θ̃, β) and a propensity parameter γ from conditional posteriors, then forms a working model for θ using estimated h(Z) and φ. The main theoretical result (Theorem 1, Section 3.3) states that the marginal posterior of θ is asymptotically normal around the oracle MLE θ̂0 at the n^{-1/2} rate, and Corollary 1.1 concludes that Bayesian credible intervals have correct frequentist coverage. The paper includes simulations and a real-data analysis of the CKD dataset, plus an extension to categorical X and a general form of the variance-weighted projection.

Significance. If the theorem and its coverage claim were fully established, the paper would provide a valuable Bayesian alternative to existing frequentist de-biasing methods for logistic regression with a binary treatment in high dimensions. The variance-weighted projection is a natural and well-motivated construction, and the use of multiple nuisance samples from posterior distributions is an interesting twist on integrated-likelihood ideas. The simulation study is thorough, covers a range of signal strengths and two sample sizes, and compares well against several existing methods. The paper also identifies a genuine technical obstacle (the failure of continuous-X orthogonality conditions when X is binary) and offers a concrete score that addresses it. The categorical extension and the general expression in Section 8 broaden the potential applicability. The central claim, however, rests on a set of assumptions whose realism and internal consistency need careful scrutiny, and the proof as written leaves several gaps that a revision would need to close.

major comments (4)

[Section 3.2, Assumption A5 (Eq. 3.5)] Assumption A5 is effectively a correct-specification assumption at a parametric rate, despite the text's claim that no correct model for P(X_i=1|Z_i) is assumed. The assumption requires the logistic working model (γ-WM) to estimate the true propensity with uniform log-odds error M√(log(d∨n)/n). If the true propensity is not exactly logistic, the posterior for γ concentrates at a pseudo-true value, and the log-odds bias Z_i^T(γ* − γ_true_logit) is generically of constant order for at least some i, violating A5. This is not a minor technicality: Lemma 3 uses A5 to control the T3 term in |h(Z_i)−h0(Z_i)|, and Lemma 7 uses the same rate to make the second-order remainder in the score expansion negligible. If A5 fails, the estimated h(Z) is asymptotically biased, the orthogonality correction does not drive E[B_n] to zero after √n scaling, and the posterior no longer centers at θ0. Consequently, the coverage guarantee in Corollary 1.1 is not supported in settings where the propensity model is misspecified. The paper's own Section 5.3 concedes that the working model '(γ-WM) might not strictly apply' in the real-data experiment, so the discrepancy between the theory and the claimed empirical validity is concrete. A revision should either (i) state explicitly that the theory assumes a correctly specified logistic propensity model and soften the 'valid inference' claim accordingly, or (ii) develop a doubly robust version that permits misspecification of one of the nuisance models.
[Section 3.3, Eq. (3.9) and proof in Appendix B (around B.16)] The definition of σ_n is inconsistent with its use. Eq. (3.9) defines σ_n = √( −∂² L_n/∂θ² |_{θ=θ̂0} ), which is O(√n) and is the square root of the observed Fisher information, not the standard deviation of θ̂0. The correct standard deviation is 1/√(−∂² L_n/∂θ²). The proof then uses (σ_n²)^{-1} = L''_n(θ̂0|h0,φ0), which has the wrong sign and is also inconsistent with (3.9). This matters because Theorem 1 and Corollary 1.1 rely on σ_n as the scaling in the normal approximation; if σ_n is mis-defined, the limiting distribution statement is not well posed. The text in Section 3.3 even says 'σn = O(√n)' immediately after defining σ_n as the standard deviation, which confirms the error. A revision must correct the definition of σ_n and ensure that all subsequent expressions (B.16), (B.20), and (B.25) are consistent with that corrected definition.
[Lemma 7 (Appendix B, Eq. B.9–B.10)] The proof of Lemma 7 invokes Lemma 2 from Belloni et al. (2013) to control the empirical process deviations in (B.10), but the observations (Y_i, X_i, Z_i) are not independent conditional on the estimated nuisance quantities h(Z) and φ, because h(Z_i) and φ_i are constructed from the same posterior sample (θ̃,β). Lemma 2 as stated applies to independent observations; the manuscript does not verify the required conditions (e.g., uniform entropy or bracketing numbers) for the dependent case. Moreover, the statement 'By Neyman orthogonality, E[∂_t b_n(η0)[η−η0]] = 0' is only true when h0 is the true variance-weighted projection; if A5 fails (see major comment above), this expectation is not zero and the claimed √n|E[b_n − b_n0]| → 0 bound in (B.9) does not follow. The proof of Lemma 7 is therefore incomplete as written.
[Theorem 1 proof (Appendix B, first paragraph and B.40)] The proof establishes the conclusion only on a high-probability set E and then conditions on a further high-probability set B. The passage from (B.14) shows that the marginal posterior is close to the conditional posterior on B, but the stochastic convergence results used to prove (B.25) and (B.27) are unconditional statements (over the data), not statements conditional on Y, X, Z. To prove that the posterior probability P(θ ∈ (θ̂0 + aσ_n, θ̂0 + bσ_n) | Y,X,Z) converges in probability to the normal CDF, one needs to show that the ratio of integrals converges uniformly over the data with high probability. The manuscript does not provide a rigorous justification connecting the unconditional convergence of the score terms to the conditional posterior probability. This gap is load-bearing because the theorem's conclusion is exactly a convergence in probability of a function of the data.

minor comments (4)

[Section 2.1, paragraph after Eq. (2.9)] The sentence 'We use the notations h0(Z) and h(Z) to represent the vectors (h0Z1),...,h0(Zn)) and (h(Z1),...,h(Zn)), respectively' contains missing parentheses and is hard to read; it should be edited for clarity.
[Section 1.3, title] The phrase 'Connections to Causal Inference' contains a typo in the body text: 'casual inference' appears in the first sentence; it should be 'causal inference'.
[Section 3.3, text before Theorem 1] The sentence 'The theorem states that the posterior concentrates around θ̂0 which is the oracle estimator (3.8) at n−1/2 rate because σn = O(√n)' is misleading: if σ_n were the standard deviation it would be O(n^{-1/2}); the sentence becomes correct only if σ_n is re-defined as the inverse standard deviation. This is a direct consequence of the Eq. (3.9) error and should be fixed.
[Section 6.2, paragraph 1] The paper states that in the real-data analysis 'we obtain the nuisance estimates via LASSO' without noting that this violates Assumption A4, which is about posterior concentration of the spike-and-slab method. A sentence explaining that this is a heuristic used for computational convenience, not covered by the theory, would be appropriate.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the variance-weighted projection is derived from an orthogonality condition, the coverage theorem is proven under explicit assumptions, and the self-citations are motivational rather than load-bearing.

full rationale

The central claim (Theorem 1 and Corollary 1.1) is that the marginal posterior for theta concentrates around the oracle MLE and yields valid credible intervals. The construction is not fitted to theta: h0(Z) is defined in Eq. (2.9) by solving the orthogonality condition E[partial_beta psi]=0 (Eq. 2.8), and the estimated h(Z) and phi are formed from separate nuisance posterior samples ((theta~,beta)-CP and gamma-CP). The proof proceeds by bounding the difference between the estimated and oracle log-likelihoods using Assumptions A4 and A5 plus Lemma 7, whose Neyman-orthogonality step is a mathematical consequence of the h0 definition, not an imported equality. Assumption A5 requires the logistic working model for the propensity to concentrate at rate sqrt(log(d∨n)/n); this is a substantive regularity condition, not a fitted parameter renamed as a prediction. Simulations use known theta0 values and an ORACLE benchmark, so coverage is checked against an external truth rather than calibrated. The self-citations to Wu et al. (2023) and Ojha and Narisetty (2023) supply the conditional-posterior framework and the spike-and-slab concentration rate, but the novel variance-weighted projection and its asymptotic normality are derived and proved in the present paper; none of these citations is invoked as a uniqueness theorem or as a substitute for the proof. The Section 5.3 caveat that the working model 'might not strictly apply' is an applicability limitation, not a circular step. I find no equation whose output is equivalent to its own input.

Assumptions & free parameters 2 free parameters · 8 assumptions · 2 invented entities

The central claim rests on high-level concentration assumptions A4 and A5 for the nuisance and propensity score posteriors, which are not derived in the paper. The variance-weighted projection is a designed quantity rather than a free parameter. The working model (theta-WM) is an ad hoc construction whose only independent check is finite-sample simulation under a correctly specified propensity model.

free parameters (2)

prior variance lambda for theta = 10
Chosen by hand in Section 4 (P3); a larger lambda makes the prior more diffuse, but it is not fitted to the outcome and does not affect the asymptotic result.
spike-and-slab hyperparameters (tau0n, tau1n, qn) = tau0n^2=1/n, n*tau1n^2=max(n, 0.01*d^2.1), qn set so P(sum Ij > K)=0.1 with K=max(10, log n)
Used for the nuisance parameters beta and gamma, following Narisetty et al. (2019); these are tuning choices, not fitted to the target effect.

assumptions (8)

domain assumption A1: log d = o(n)
Dimension growth condition stated in Section 3.2.
domain assumption A2: nuisance covariates bounded
Boundedness of |Z_ij|; needed for concentration inequalities in proofs.
domain assumption A3: sparsity s^2 log(d or n) = o(sqrt(n))
Sparsity condition for the true nuisance parameter beta0.
domain assumption A4: posterior concentration of (theta~, beta)
Assumed, not proved for the specific spike-and-slab implementation; it is a high-level concentration condition.
domain assumption A5: propensity score estimator concentration
Assumes the logistic working model estimator for P(X_i=1|Z_i) is uniformly close to the true propensity score at rate sqrt(log(d or n)/n), effectively a correct-specification condition that the paper claims not to make.
domain assumption A6: prior on theta continuous and positive at theta0, theta0 bounded
Standard regularity condition on the prior and true parameter.
ad hoc to paper Working model (theta-WM) and oracle equivalence
The conditional posterior for theta is defined through a working model that is not a standard Bayesian posterior; the oracle identity in Section 2.2.6 is required for the proof.
standard math Pólya-Gamma representation of logistic likelihood
Used in the Gibbs sampler (Appendix A); standard latent-variable representation.

invented entities (2)

Variance-weighted projection h0(Z_i) independent evidence
purpose: Weighted conditional expectation of X_i used to define an orthogonal score for binary treatment X_i.
Defined in (2.9) and generalized in (8.3) by solving the conditional moment condition E[(X_i-h0(Z_i)) mu0_i (1-mu0_i)|Z_i]=0. It is a mathematical construction with empirical consequences (coverage of intervals), not a physical entity.
Working model (theta-WM)
purpose: Provides a likelihood for theta with the sampled nuisance quantities (h(Z), phi) treated as fixed offsets.
Introduced ad hoc to enable sampling of theta; its validity is only justified by the oracle equivalence and the theorem's assumptions, not by independent evidence.

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Cite this review

Pith. "Pith review of Valid Bayesian Inference based on Variance Weighted Projection for High-Dimensional Logistic Regression with Binary Covariates." pith.science (2026). https://pith.science/paper/MI3OO573

@misc{pith2026241117618,
  author       = {Pith},
  title        = {Pith review of: Valid Bayesian Inference based on Variance Weighted Projection for High-Dimensional Logistic Regression with Binary Covariates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MI3OO573}},
  note         = {Machine review of arXiv:2411.17618}
}

read the original abstract

We address the challenge of conducting inference for a categorical treatment effect related to a binary outcome variable while taking into account high-dimensional baseline covariates. The conventional technique used to establish orthogonality for the treatment effect from nuisance variables in continuous cases is inapplicable in the context of binary treatment. To overcome this obstacle, an orthogonal score tailored specifically to this scenario is formulated which is based on a variance-weighted projection. Additionally, a novel Bayesian framework is proposed to facilitate valid inference for the desired low-dimensional parameter within the complex framework of high-dimensional logistic regression. We provide uniform convergence results, affirming the validity of credible intervals derived from the posterior distribution. The effectiveness of the proposed method is demonstrated through comprehensive simulation studies and real data analysis.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Causal in ference in high di- mensions: a marriage between Bayesian modeling and good frequent ist properties,

Agresti, A. (2012), Categorical data analysis , vol. 792, John Wiley & Sons. 3 Antonelli, J., Papadogeorgou, G., and Dominici, F. (2022), “Causal in ference in high di- mensions: a marriage between Bayesian modeling and good frequent ist properties,” Bio- metrics, 78, 100–114. 8 Armagan, A., Dunson, D. B., and Lee, J. (2013), “Generalized doub le Pareto s...

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Let |hi(t)| ≤ |tTWi|, K/4 > ¯σ2 = sup t∈T ¯E[hi(t)2ξ2 i ], ˜p = dim(Wi), and ‖T ‖1 = sup t∈T ‖t‖1

Consider the notations En(ri) = 1 n ∑ n i=1ri and ¯E(ri) = 1 n ∑ n i=1E[ri]. Let |hi(t)| ≤ |tTWi|, K/4 > ¯σ2 = sup t∈T ¯E[hi(t)2ξ2 i ], ˜p = dim(Wi), and ‖T ‖1 = sup t∈T ‖t‖1. We have P ( sup t∈T |(En − ¯E)[hi(t)ξi]|> K‖T ‖1 √ M√n ) ≤ 32˜p exp (−K 2 16 ) + P ( max j≤˜p En[W 2 ijξ2 i ]>M ) . Lemma 2 has been proved in ( Belloni et al. , 2013). Lemma

work page 2013

[3] [5]

Large-scale Baye sian logistic regression for text categorization,

6 41 Genkin, A., Lewis, D. D., and Madigan, D. (2007), “Large-scale Baye sian logistic regression for text categorization,” Technometrics, 49, 291–304. 6 George, E. I. and McCulloch, R. E. (1993), “Variable selection via Gib bs sampling,” Journal of the American Statistical Association , 88, 881–889. 6 Ghosh, J., Li, Y., and Mitra, R. (2018), “On the use ...

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A version of the results in this lemma have been proved in ( Belloni et al. , 2013). We will use second order Taylor’s series expansion with directional derivatives. The directional derivative of f at ˜x in the direction of [ x −x0] is deﬁned as: ∂tf (˜x)[x −x0] = lim t→0 f (˜x +t[x −x0]) −f (˜x) t . Similarly, the second directional derivative can be deﬁ...

work page 2013

[5] [12]

Lemma 1 consists of preliminary results shown in ( Walker, 1969; Schervish, 2012)

= OP (1). Lemma 1 consists of preliminary results shown in ( Walker, 1969; Schervish, 2012). Lemma

work page 1969

[6] [15]

(B.22) From Walker (1969), we already know how to control the ˆθ0 related term

(B.21) Let’s try to understand the behavior of the following integral (for 0<ǫ< 1): Iǫ = ∫ N0(δ) exp(Ln(θ | h(Z),φ ) − Ln(θ | h0(Z),φ 0)) exp [ −(θ − ˆθ0)2(1 ±ǫ) 2σ2 n ] dθ. (B.22) From Walker (1969), we already know how to control the ˆθ0 related term. Because of high dimensional setting, we have new term to deal with. Note that w e can express the diﬀer...

work page 1969

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Does maternal birth out come diﬀerentially inﬂu- ence the occurrence of infant death among African Americans and European Americans?

28 Masho, S. W. and Archer, P. W. (2011), “Does maternal birth out come diﬀerentially inﬂu- ence the occurrence of infant death among African Americans and European Americans?” Maternal and child health journal , 15, 1249–1256. 28, 29 Narisetty, N. N. and He, X. (2014), “Bayesian variable selection wit h shrinking and diﬀusing priors,” The Annals of Stati...

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Eﬀects of model selection on inferenc e,

49 P¨ otscher, B. M. (1991), “Eﬀects of model selection on inferenc e,” Econometric Theory, 7, 163–185. 7 — (2009), “Conﬁdence sets based on sparse estimators are nece ssarily large,” Sankhy¯ a: The Indian Journal of Statistics, Series A (2008-) , 1–18. 7 P¨ otscher, B. M. and Leeb, H. (2009), “On the distribution of penalized maximum likelihood estimator...

work page 1991