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Model selection is the process of choosing from a class of candidate models given data. For instance, methods such as the LASSO and sparse identification of nonlinear dynamics (SINDy) formulate model selection as finding a sparse solution to a linear system of equations determined by training data. However, absent strong assumptions, such methods are highly unstable: if a single data point is removed from the training set, a different model may be selected. This paper presents a new approach to stabilizing model selection that leverages a combination of bagging and an "inflated" argmax operation. Our method selects a small collection of models that all fit the data, and it is stable in that, with high probability, the removal of any training point will result in a collection of selected models that overlaps with the original collection. In addition to developing theoretical guarantees, we illustrate this method in (a) a simulation in which strongly correlated covariates make standard LASSO model selection highly unstable and (b) a Lotka-Volterra model selection problem focused on identifying how competition in an ecosystem influences species' abundances. In both settings, the proposed method yields stable and compact collections of selected models, outperforming a variety of benchmarks.

We propose a new framework for algorithmic stability in the context of multiclass classification. In practice, classification algorithms often operate by first assigning a continuous score (for instance, an estimated probability) to each possible label, then taking the maximizer -- i.e., selecting the class that has the highest score. A drawback of this type of approach is that it is inherently unstable, meaning that it is very sensitive to slight perturbations of the training data, since taking the maximizer is discontinuous. Motivated by this challenge, we propose a pipeline for constructing stable classifiers from data, using bagging (i.e., resampling and averaging) to produce stable continuous scores, and then using a stable relaxation of argmax, which we call the "inflated argmax," to convert these scores to a set of candidate labels. The resulting stability guarantee places no distributional assumptions on the data, does not depend on the number of classes or dimensionality of the covariates, and holds for any base classifier. Using a common benchmark data set, we demonstrate that the inflated argmax provides necessary protection against unstable classifiers, without loss of accuracy.

Model averaging techniques based on resampling methods (such as bootstrapping or subsampling) have been utilized across many areas of statistics, often with the explicit goal of promoting stability in the resulting output. We provide a general, finite-sample theoretical result guaranteeing the stability of bagging when applied to algorithms that return outputs in a general space, so that the output is not necessarily a real-valued -- for example, an algorithm that estimates a vector of weights or a density function. We empirically assess the stability of bagging on synthetic and real-world data for a range of problem settings, including causal inference, nonparametric regression, and Bayesian model selection.

Contemporary scientific research is a distributed, collaborative endeavor, carried out by teams of researchers, regulatory institutions, funding agencies, commercial partners, and scientific bodies, all interacting with each other and facing different incentives. To maintain scientific rigor, statistical methods should acknowledge this state of affairs. To this end, we study hypothesis testing when there is an agent (e.g., a researcher or a pharmaceutical company) with a private prior about an unknown parameter and a principal (e.g., a policymaker or regulator) who wishes to make decisions based on the parameter value. The agent chooses whether to run a statistical trial based on their private prior and then the result of the trial is used by the principal to reach a decision. We show how the principal can conduct statistical inference that leverages the information that is revealed by an agent's strategic behavior -- their choice to run a trial or not. In particular, we show how the principal can design a policy to elucidate partial information about the agent's private prior beliefs and use this to control the posterior probability of the null. One implication is a simple guideline for the choice of significance threshold in clinical trials: the type-I error level should be set to be strictly less than the cost of the trial divided by the firm's profit if the trial is successful.

Bagging is an important technique for stabilizing machine learning models. In this paper, we derive a finite-sample guarantee on the stability of bagging for any model. Our result places no assumptions on the distribution of the data, on the properties of the base algorithm, or on the dimensionality of the covariates. Our guarantee applies to many variants of bagging and is optimal up to a constant. Empirical results validate our findings, showing that bagging successfully stabilizes even highly unstable base algorithms.

Despite the popularity of the false discovery rate (FDR) as an error control metric for large-scale multiple testing, its close Bayesian counterpart the local false discovery rate (lfdr), defined as the posterior probability that a particular null hypothesis is false, is a more directly relevant standard for justifying and interpreting individual rejections. However, the lfdr is difficult to work with in small samples, as the prior distribution is typically unknown. We propose a simple multiple testing procedure and prove that it controls the expectation of the maximum lfdr across all rejections; equivalently, it controls the probability that the rejection with the largest p-value is a false discovery. Our method operates without knowledge of the prior, assuming only that the p-value density is uniform under the null and decreasing under the alternative. We also show that our method asymptotically implements the oracle Bayes procedure for a weighted classification risk, optimally trading off between false positives and false negatives. We derive the limiting distribution of the attained maximum lfdr over the rejections, and the limiting empirical Bayes regret relative to the oracle procedure.

Consider the relationship between a regulator (the principal) and an experimenter (the agent) such as a pharmaceutical company. The pharmaceutical company wishes to sell a drug for profit, whereas the regulator wishes to allow only efficacious drugs to be marketed. The efficacy of the drug is not known to the regulator, so the pharmaceutical company must run a costly trial to prove efficacy to the regulator. Critically, the statistical protocol used to establish efficacy affects the behavior of a strategic, self-interested agent; a lower standard of statistical evidence incentivizes the agent to run more trials that are less likely to be effective. The interaction between the statistical protocol and the incentives of the pharmaceutical company is crucial for understanding this system and designing protocols with high social utility. In this work, we discuss how the regulator can set up a protocol with payoffs based on statistical evidence. We show how to design protocols that are robust to an agent's strategic actions, and derive the optimal protocol in the presence of strategic entrants.

Multivariate, heteroscedastic errors complicate statistical inference in many large-scale denoising problems. Empirical Bayes is attractive in such settings, but standard parametric approaches rest on assumptions about the form of the prior distribution which can be hard to justify and which introduce unnecessary tuning parameters. We extend the nonparametric maximum likelihood estimator (NPMLE) for Gaussian location mixture densities to allow for multivariate, heteroscedastic errors. NPMLEs estimate an arbitrary prior by solving an infinite-dimensional, convex optimization problem; we show that this convex optimization problem can be tractably approximated by a finite-dimensional version. The empirical Bayes posterior means based on an NPMLE have low regret, meaning they closely target the oracle posterior means one would compute with the true prior in hand. We prove an oracle inequality implying that the empirical Bayes estimator performs at nearly the optimal level (up to logarithmic factors) for denoising without prior knowledge. We provide finite-sample bounds on the average Hellinger accuracy of an NPMLE for estimating the marginal densities of the observations. We also demonstrate the adaptive and nearly-optimal properties of NPMLEs for deconvolution. We apply our method to two denoising problems in astronomy, constructing a fully data-driven color-magnitude diagram of 1.4 million stars in the Milky Way and investigating the distribution of 19 chemical abundance ratios for 27 thousand stars in the red clump. We also apply our method to hierarchical linear models, illustrating the advantages of nonparametric shrinkage of regression coefficients on an education data set and on a microarray data set.

We study the problem of high-dimensional covariance estimation under the constraint that the partial correlations are nonnegative. The sign constraints dramatically simplify estimation: the Gaussian maximum likelihood estimator is well defined with only two observations regardless of the number of variables. We analyze its performance in the setting where the dimension may be much larger than the sample size. We establish that the estimator is both high-dimensionally consistent and minimax optimal in the symmetrized Stein loss. We also prove a negative result which shows that the sign-constraints can introduce substantial bias for estimating the top eigenvalue of the covariance matrix.

It is well known that the isotonic least squares estimator is characterized as the derivative of the greatest convex minorant of a random walk. Provided the walk has exchangeable increments, we prove that the slopes of the greatest convex minorant are distributed as order statistics of the running averages. This result implies an exact non-asymptotic formula for the squared error risk of least squares in isotonic regression when the true sequence is constant that holds for every exchangeable error distribution.

Through mathematical analysis and simulations, online ratings and their impact on businesses are characterized through two parameters: an inherent and objective restaurant quality factor, and the accuracy of customers' gut feeling about a business. Within this model, it is found that online ratings are seldom accurate mainly because of the low or high accuracy in customers' gut feelings.