Exact adaptive confidence intervals for linear regression coefficients

Hoff, Peter D.; Yu, Chaoyu

doi:10.1214/18-ejs1517

Cited by 14 publications

(8 citation statements)

References 22 publications

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“…Intuitively, this provides moment-based estimates of τ 2 and σ 2 by minimizing a loss function relating the empirical variance yy to the variance XX τ 2 + σ 2 I n under the model (1), while requiring positive definiteness of XX τ 2 + σ 2 I n . Hoff and Yu (2017) demonstrate that these estimates will be consistent for τ 2 and σ 2 as n and p → ∞ even if the distribution of β is not normal. Solving (7) has been treated thoroughly in the random effects literature (Demidenko, 2013).…”

Section: Estimation Of σ 2 and τmentioning

confidence: 74%

Testing Sparsity-Inducing Penalties

Griffin

Hoff

2019

Journal of Computational and Graphical Statistics

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Many penalized maximum likelihood estimators correspond to posterior mode estimators under specific prior distributions. Appropriateness of a particular class of penalty functions can therefore be interpreted as the appropriateness of a prior for the parameters. For example, the appropriateness of a lasso penalty for regression coefficients depends on the extent to which the empirical distribution of the regression coefficients resembles a Laplace distribution. We give a testing procedure of whether or not a Laplace prior is appropriate and accordingly, whether or not using a lasso penalized estimate is appropriate. This testing procedure is designed to have power against exponential power priors which correspond to q penalties. Via simulations, we show that this testing procedure achieves the desired level and has enough power to detect violations of the Laplace assumption when the numbers of observations and unknown regression coefficients are large. We then introduce an adaptive procedure that chooses a more appropriate prior and corresponding penalty from the class of exponential power priors when the null hypothesis is rejected. We show that this can improve estimation of the regression coefficients both when they are drawn from an exponential power distribution and when they are drawn from a spike-and-slab distribution.

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Section: Estimation Of σ 2 and τmentioning

confidence: 74%

Testing Sparsity-Inducing Penalties

Griffin

Hoff

2019

Journal of Computational and Graphical Statistics

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“…Outside of decision theory, Bayesian models have played a role in the construction of smaller confidence intervals with exact frequentist coverage, initially by Pratt (1963) and more recently in the context of empirical Bayes inference by Hoff & Yu (2019) and post-selection inference by Woody et al (2020). In the context of hypothesis testing, Hoff (2020) leveraged a similar approach to increase power across multiple hypotheses while maintaining exact coverage.…”

Section: Additional Related Workmentioning

confidence: 99%

Confidently Comparing Estimators with the c-value

Trippe¹,

Deshpande²,

Broderick³

2021

Preprint

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Modern statistics provides an ever-expanding toolkit for estimating unknown parameters. Consequently, applied statisticians frequently face a difficult decision: retain a parameter estimate from a familiar method or replace it with an estimate from a newer or complex one. While it is traditional to compare estimators using risk, such comparisons are rarely conclusive in realistic settings.In response, we propose the "c-value" as a measure of confidence that a new estimate achieves smaller loss than an old estimate on a given dataset. We show that it is unlikely that a computed c-value is large and that the new estimate has larger loss than the old. Therefore, just as a small p-value provides evidence to reject a null hypothesis, a large c-value provides evidence to use a new estimate in place of the old. For a wide class of problems and estimators, we show how to compute a c-value by first constructing a data-dependent high-probability lower bound on the difference in loss. The c-value is frequentist in nature, but we show that it can provide a validation of Bayesian estimates in real data applications involving hierarchical models and Gaussian processes.

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“…Our work builds directly off the "frequentist assisted by Bayes" (FAB) framework, which was articulated in its earliest form by Pratt (1963) and was then rediscovered and substantially extended by Yu and Hoff (2018). This technique was originally proposed for constructing confidence intervals for group-level means in hierarchical normal models, and has also been extended to constructing confidence intervals for coefficients in a linear regression by Hoff and Yu (2017). First we will summarize this framework as introduced for nonselective inference, and next we will generalize it to the case of post-selection inference.…”

Section: "Frequentist Assisted By Bayes" (Fab) Inferencementioning

confidence: 99%

Optimal post-selection inference for sparse signals: a nonparametric empirical-Bayes approach

Woody,

Padilla,

Scott

2018

Preprint

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A large body of recent Bayesian work has focused on the question of how to find sparse signals. Much less work, however, has been done on the natural follow-up question: how to make valid inferences for the magnitude of those signals once they've been found. Ordinary Bayesian credible intervals are not necessarily appropriate for this task: in many circumstances, they suffer from selection bias, owing to the fact that the target of inference is chosen adaptively. There are many purely frequentist proposals for addressing this problem. But these typically require sacrificing the benefits of shrinkage or "borrowing strength" inherent to Bayesian modeling, resulting in confidence intervals that are needlessly wide. On the flip side, there are also Bayesian proposals for addressing this problem, most notably that of Yekutieli ( 2012), who constructs selectionadjusted posterior distributions. The resulting credible intervals, however, have poor frequentist performance: for nearly all values of the underlying parameter, they fail to exhibit the correct nominal coverage. Thus there is an unmet need for approaches to inference that correctly adjust for selection, and incorporate the benefits of shrinkage while maintaining exact frequentist coverage.We address this gap by proposing a nonparametric empirical-Bayes approach for constructing optimal selection-adjusted confidence sets. The method produces confidence sets that are as short as possible, while both adjusting for selection and maintaining exact frequentist coverage uniformly across the whole parameter space. Across a series of examples, the method outperforms existing frequentist techniques for post-selection inference, producing confidence sets that are notably shorter but with the same coverage guarantee.

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Exact adaptive confidence intervals for linear regression coefficients

Cited by 14 publications

References 22 publications

Testing Sparsity-Inducing Penalties

Testing Sparsity-Inducing Penalties

Confidently Comparing Estimators with the c-value

Optimal post-selection inference for sparse signals: a nonparametric empirical-Bayes approach

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