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Hannes Leeb

Publications and source records attributed to Hannes Leeb.

At least 19 recordsLinked to original sources
arXiv2025

An identity involving counts of binary matrices

In the context of generating uniform random contingency tables with pre-specified marginals, the number of (binary) matrices with given row- and column-sums is a well-studied object in the literature. We will denote this number by $N(p,q)$, where $p$ and $q$ are the vectors of row- and column-sums. The existing literature is mainly focused on computing or approximating $N(p,q)$. In this paper, we present two identities for polynomials whose coefficients depend on the $N(p,q)$ and explore some consequences.

math.CO↗
arXiv2024

A (tight) upper bound for the length of confidence intervals with conditional coverage

We show that two popular selective inference procedures, namely data carving (Fithian et al., 2017) and selection with a randomized response (Tian et al., 2018b), when combined with the polyhedral method (Lee et al., 2016), result in confidence intervals whose length is bounded. This contrasts results for confidence intervals based on the polyhedral method alone, whose expected length is typically infinite (Kivaranovic and Leeb, 2020). Moreover, we show that these two procedures always dominate corresponding sample-splitting methods in terms of interval length.

stat.ME↗
arXiv2023

Uncertainty quantification via cross-validation and its variants under algorithmic stability

Recently, there has been substantial interest in statistical guarantees for cross-validation (CV) methods of uncertainty quantification in statistical learning (cf. Barber et al. 2021a, Liang and Barber 2024, Steinberger and Leeb 2023). These guarantees should hold under minimal assumptions on the data generating process and conditional on the training data, because numerous predictions are usually computed based on one and the same training sample. We push this objective to the limit: We prove asymptotic conditional conservativeness of CV, that is, the probability of the actual coverage probability, conditional on the training data, undershooting its nominal level vanishes asymptotically, under minimal assumptions. In particular, we impose a stability condition, require that the prediction error is stochastically bounded, and show that neither condition can be dropped in general. By way of an asymptotic equivalence result, we also show that the closely related CV+ method of Barber et al. (2021a) provides exactly the same conditional statistical guarantees as CV in large samples, thereby extending the range of applicability of CV+ to the high-dimensional regime. We conclude that, in view of its marginal coverage guarantee, CV+ does indeed improve over simple CV. For our proofs we introduce a new concept called L\'evy gauge, which can be of independent interest.

math.ST↗
arXiv2022

Conditional predictive inference for stable algorithms

We investigate generically applicable and intuitively appealing prediction intervals based on $k$-fold cross validation. We focus on the conditional coverage probability of the proposed intervals, given the observations in the training sample (hence, training conditional validity), and show that it is close to the nominal level, in an appropriate sense, provided that the underlying algorithm used for computing point predictions is sufficiently stable when feature-response pairs are omitted. Our results are based on a finite sample analysis of the empirical distribution function of $k$-fold cross validation residuals and hold in non-parametric settings with only minimal assumptions on the error distribution. To illustrate our results, we also apply them to high-dimensional linear predictors, where we obtain uniform asymptotic training conditional validity as both sample size and dimension tend to infinity at the same rate and consistent parameter estimation typically fails. These results show that despite the serious problems of resampling procedures for inference on the unknown parameters (cf. Bickel and Freedman, 1983; El Karoui and Purdom, 2018; Mammen, 1996), cross validation methods can be successfully applied to obtain reliable predictive inference even in high dimensions and conditionally on the training data.

math.ST↗
arXiv2020

A Tale of Two Referees

Success in academia hinges on publishing in top tier journals. This requires innovative results. And this requires clear and convincing presentation of said results. Presentation can make the difference of one tier in journal level. A lot of useful advice on this topic is available online from well-respected outlets; see, for example, El-Omar (2014); Gould (2014); Neiles et al. (2015); Notz and Kafadar (2011); or Sachdeva (2020). This text provides a different angle.

cs.DL↗
arXiv2020

Conformal prediction intervals for the individual treatment effect

We propose several prediction intervals procedures for the individual treatment effect with either finite-sample or asymptotic coverage guarantee in a non-parametric regression setting, where non-linear regression functions, heteroskedasticity and non-Gaussianity are allowed. The construct the prediction intervals we use the conformal method of Vovk et al. (2005). In extensive simulations, we compare the coverage probability and interval length of our prediction interval procedures. We demonstrate that complex learning algorithms, such as neural networks, can lead to narrower prediction intervals than simple algorithms, such as linear regression, if the sample size is large enough.

stat.ME↗
arXiv2020

Adaptive, Distribution-Free Prediction Intervals for Deep Networks

The machine learning literature contains several constructions for prediction intervals that are intuitively reasonable but ultimately ad-hoc in that they do not come with provable performance guarantees. We present methods from the statistics literature that can be used efficiently with neural networks under minimal assumptions with guaranteed performance. We propose a neural network that outputs three values instead of a single point estimate and optimizes a loss function motivated by the standard quantile regression loss. We provide two prediction interval methods with finite sample coverage guarantees solely under the assumption that the observations are independent and identically distributed. The first method leverages the conformal inference framework and provides average coverage. The second method provides a new, stronger guarantee by conditioning on the observed data. Lastly, our loss function does not compromise the predictive accuracy of the network like other prediction interval methods. We demonstrate the ease of use of our procedures as well as its improvements over other methods on both simulated and real data. As most deep networks can easily be modified by our method to output predictions with valid prediction intervals, its use should become standard practice, much like reporting standard errors along with mean estimates.

stat.ML↗
arXiv2019

On the length of post-model-selection confidence intervals conditional on polyhedral constraints

Valid inference after model selection is currently a very active area of research. The polyhedral method, pioneered by Lee, et al. (2016), allows for valid inference after model selection if the model selection event can be described by polyhedral constraints. In that reference, the method is exemplified by constructing two valid confidence intervals when the Lasso estimator is used to select a model. We here study the length of these intervals. For one of these confidence intervals, which is easier to compute, we find that its expected length is always infinite. For the other of these confidence intervals, whose computation is more demanding, we give a necessary and sufficient condition for its expected length to be infinite. In simulations, we find that this sufficient condition is typically satisfied, unless the selected model includes almost all or almost none of the available regressors. For the distribution of confidence interval length, we find that the $κ$-quantiles behave like $1/(1-κ)$ for $κ$ close to $1$. Our results can also be used to analyze other confidence intervals that are based on the polyhedral method.

math.ST↗
arXiv2019

Statistical inference with F-statistics when fitting simple models to high-dimensional data

We study linear subset regression in the context of the high-dimensional overall model $y = \vartheta+θ' z + ε$ with univariate response $y$ and a $d$-vector of random regressors $z$, independent of $ε$. Here, "high-dimensional" means that the number $d$ of available explanatory variables is much larger than the number $n$ of observations. We consider simple linear sub-models where $y$ is regressed on a set of $p$ regressors given by $x = M'z$, for some $d \times p$ matrix $M$ of full rank $p < n$. The corresponding simple model, i.e., $y=α+β' x + e$, can be justified by imposing appropriate restrictions on the unknown parameter $θ$ in the overall model; otherwise, this simple model can be grossly misspecified. In this paper, we establish asymptotic validity of the standard $F$-test on the surrogate parameter $β$, in an appropriate sense, even when the simple model is misspecified.

math.ST↗
arXiv2018

Admissibility of the usual confidence set for the mean of a univariate or bivariate normal population: The unknown-variance case

In the Gaussian linear regression model (with unknown mean and variance), we show that the standard confidence set for one or two regression coefficients is admissible in the sense of Joshi (1969). This solves a long-standing open problem in mathematical statistics, and this has important implications on the performance of modern inference procedures post-model-selection or post-shrinkage, particularly in situations where the number of parameters is larger than the sample size. As a technical contribution of independent interest, we introduce a new class of conjugate priors for the Gaussian location-scale model.

math.ST↗
arXiv2018

Prediction out-of-sample using block shrinkage estimators: model selection and predictive inference

In a linear regression model with random design, we consider a family of candidate models from which we want to select a `good' model for prediction out-of-sample. We fit the models using block shrinkage estimators, and we focus on the challenging situation where the number of explanatory variables can be of the same order as sample size and where the number of candidate models can be much larger than sample size. We develop an estimator for the out-of-sample predictive performance, and we show that the empirically best model is asymptotically as good as the truly best model. Using the estimator corresponding to the empirically best model, we construct a prediction interval that is approximately valid and short with high probability, i.e., we show that the actual coverage probability is close to the nominal one and that the length of this prediction interval is close to the length of the shortest but infeasible prediction interval. All results hold uniformly over a large class of data-generating processes. These findings extend results of Leeb (2009), where the models are fit using least-squares estimators, and of Huber (2013), where the models are fit using shrinkage estimators without block structure.

math.ST↗
arXiv2017

Valid confidence intervals for post-model-selection predictors

We consider inference post-model-selection in linear regression. In this setting, Berk et al.(2013) recently introduced a class of confidence sets, the so-called PoSI intervals, that cover a certain non-standard quantity of interest with a user-specified minimal coverage probability, irrespective of the model selection procedure that is being used. In this paper, we generalize the PoSI intervals to post-model-selection predictors.

math.ST↗
arXiv2016

On conditional moments of high-dimensional random vectors given lower-dimensional projections

One of the most widely used properties of the multivariate Gaussian distribution, besides its tail behavior, is the fact that conditional means are linear and that conditional variances are constant. We here show that this property is also shared, in an approximate sense, by a large class of non-Gaussian distributions. We allow for several conditioning variables and we provide explicit non-asymptotic results, whereby we extend earlier findings of Hall and Li (1993) and Leeb (2013).

math.ST↗
arXiv2016

Leave-one-out prediction intervals in linear regression models with many variables

We study prediction intervals based on leave-one-out residuals in a linear regression model where the number of explanatory variables can be large compared to sample size. We establish uniform asymptotic validity (conditional on the training sample) of the proposed interval under minimal assumptions on the unknown error distribution and the high dimensional design. Our intervals are generic in the sense that they are valid for a large class of linear predictors used to obtain a point forecast, such as robust M-estimators, James-Stein type estimators and penalized estimators like the LASSO. These results show that despite the serious problems of resampling procedures for inference on the unknown parameters, leave-one-out methods can be successfully applied to obtain reliable predictive inference even in high dimensions.

math.ST↗
arXiv2015

On Various Confidence Intervals Post-Model-Selection

We compare several confidence intervals after model selection in the setting recently studied by Berk et al. [Ann. Statist. 41 (2013) 802-837], where the goal is to cover not the true parameter but a certain nonstandard quantity of interest that depends on the selected model. In particular, we compare the PoSI-intervals that are proposed in that reference with the "naive" confidence interval, which is constructed as if the selected model were correct and fixed a priori (thus ignoring the presence of model selection). Overall, we find that the actual coverage probabilities of all these intervals deviate only moderately from the desired nominal coverage probability. This finding is in stark contrast to several papers in the existing literature, where the goal is to cover the true parameter.

math.ST↗
arXiv2014

Testing in the Presence of Nuisance Parameters: Some Comments on Tests Post-Model-Selection and Random Critical Values

We point out that the ideas underlying some test procedures recently proposed for testing post-model-selection (and for some other test problems) in the econometrics literature have been around for quite some time in the statistics literature. We also sharpen some of these results in the statistics literature. Furthermore, we show that some intuitively appealing testing procedures, that have found their way into the econometrics literature, lead to tests that do not have desirable size properties, not even asymptotically.

math.ST↗
arXiv2013

Shrinkage estimators for prediction out-of-sample: Conditional performance

We find that, in a linear model, the James-Stein estimator, which dominates the maximum-likelihood estimator in terms of its in-sample prediction error, can perform poorly compared to the maximum-likelihood estimator in out-of-sample prediction. We give a detailed analysis of this phenomenon and discuss its implications. When evaluating the predictive performance of estimators, we treat the regressor matrix in the training data as fixed, i.e., we condition on the design variables. Our findings contrast those obtained by Baranchik (1973, Ann. Stat. 1:312-321) and, more recently, by Dicker (2012, arXiv:1102.2952) in an unconditional performance evaluation.

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