arXiv subjects

Ryan Martin

Publications and source records attributed to Ryan Martin.

At least 73 records · Page 4Linked to original sources

arXiv2019

Generalized inferential models for censored data

Inferential challenges that arise when data are censored have been extensively studied under the classical frameworks. In this paper, we provide an alternative generalized inferential model approach whose output is a data-dependent plausibility function. This construction is driven by an association between the distribution of the relative likelihood function at the interest parameter and an unobserved auxiliary variable. The plausibility function emerges from the distribution of a suitably calibrated random set designed to predict that unobserved auxiliary variable. The evaluation of this plausibility function requires a novel use of the classical Kaplan--Meier estimator to estimate the censoring rather than the event distribution. We prove that the proposed method provides valid inference, at least approximately, and our real- and simulated-data examples demonstrate its superior performance compared to existing methods.

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arXiv2019

Bayesian test of normality versus a Dirichlet process mixture alternative

We propose a Bayesian test of normality for univariate or multivariate data against alternative nonparametric models characterized by Dirichlet process mixture distributions. The alternative models are based on the principles of embedding and predictive matching. They can be interpreted to offer random granulation of a normal distribution into a mixture of normals with mixture components occupying a smaller volume the farther they are from the distribution center. A scalar parametrization based on latent clustering is used to cover an entire spectrum of separation between the normal distributions and the alternative models. An efficient sequential importance sampler is developed to calculate Bayes factors. Simulations indicate the proposed test can detect non-normality without favoring the nonparametric alternative when normality holds.

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arXiv2019

Empirical priors and coverage of posterior credible sets in a sparse normal mean model

Bayesian methods provide a natural means for uncertainty quantification, that is, credible sets can be easily obtained from the posterior distribution. But is this uncertainty quantification valid in the sense that the posterior credible sets attain the nominal frequentist coverage probability? This paper investigates the frequentist validity of posterior uncertainty quantification based on a class of empirical priors in the sparse normal mean model. In particular, we show that our marginal posterior credible intervals achieve the nominal frequentist coverage probability under conditions slightly weaker than needed for selection consistency and a Bernstein--von Mises theorem for the full posterior, and numerical investigations suggest that our empirical Bayes method has superior frequentist coverage probability properties compared to other fully Bayes methods.

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arXiv2019

Satellite conjunction analysis and the false confidence theorem

Satellite conjunction analysis is the assessment of collision risk during a close encounter between a satellite and another object in orbit. A counterintuitive phenomenon has emerged in the conjunction analysis literature, namely, probability dilution, in which lower quality data paradoxically appear to reduce the risk of collision. We show that probability dilution is a symptom of a fundamental deficiency in probabilistic representations of statistical inference, in which there are propositions that will consistently be assigned a high degree of belief, regardless of whether or not they are true. We call this deficiency false confidence. In satellite conjunction analysis, it results in a severe and persistent underestimate of collision risk exposure. We introduce the Martin--Liu validity criterion as a benchmark by which to identify statistical methods that are free from false confidence. Such inferences will necessarily be non-probabilistic. In satellite conjunction analysis, we show that uncertainty ellipsoids satisfy the validity criterion. Performing collision avoidance maneuvers based on ellipsoid overlap will ensure that collision risk is capped at the user-specified level. Further, this investigation into satellite conjunction analysis provides a template for recognizing and resolving false confidence issues as they occur in other problems of statistical inference.

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arXiv2019

False confidence, non-additive beliefs, and valid statistical inference

Statistics has made tremendous advances since the times of Fisher, Neyman, Jeffreys, and others, but the fundamental and practically relevant questions about probability and inference that puzzled our founding fathers remain unanswered. To bridge this gap, I propose to look beyond the two dominating schools of thought and ask the following three questions: what do scientists need out of statistics, do the existing frameworks meet these needs, and, if not, how to fill the void? To the first question, I contend that scientists seek to convert their data, posited statistical model, etc., into calibrated degrees of belief about quantities of interest. To the second question, I argue that any framework that returns additive beliefs, i.e., probabilities, necessarily suffers from {\em false confidence}---certain false hypotheses tend to be assigned high probability---and, therefore, risks systematic bias. This reveals the fundamental importance of {\em non-additive beliefs} in the context of statistical inference. But non-additivity alone is not enough so, to the third question, I offer a sufficient condition, called {\em validity}, for avoiding false confidence, and present a framework, based on random sets and belief functions, that provably meets this condition. Finally, I discuss characterizations of p-values and confidence intervals in terms of valid non-additive beliefs, which imply that users of these classical procedures are already following the proposed framework without knowing it.

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arXiv2019

Model-free posterior inference on the area under the receiver operating characteristic curve

The area under the receiver operating characteristic curve (AUC) serves as a summary of a binary classifier's performance. Methods for estimating the AUC have been developed under a binormality assumption which restricts the distribution of the score produced by the classifier. However, this assumption introduces an infinite-dimensional nuisance parameter and can be inappropriate, especially in the context of machine learning. This motivates us to adopt a model-free Gibbs posterior distribution for the AUC. We present the asymptotic Gibbs posterior concentration rate, and a strategy for tuning the learning rate so that the corresponding credible intervals achieve the nominal frequentist coverage probability. Simulation experiments and a real data analysis demonstrate the Gibbs posterior's strong performance compared to existing methods based on a rank likelihood.

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arXiv2019

Permutation-based uncertainty quantification about a mixing distribution

Nonparametric estimation of a mixing distribution based on data coming from a mixture model is a challenging problem. Beyond estimation, there is interest in uncertainty quantification, e.g., confidence intervals for features of the mixing distribution. This paper focuses on estimation via the predictive recursion algorithm, and here we take advantage of this estimator's seemingly undesirable dependence on the data ordering to obtain a permutation-based approximation of the sampling distribution which can be used to quantify uncertainty. Theoretical and numerical results confirm that the proposed method leads to valid confidence intervals, at least approximately.

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arXiv2019

Variational approximations using Fisher divergence

Modern applications of Bayesian inference involve models that are sufficiently complex that the corresponding posterior distributions are intractable and must be approximated. The most common approximation is based on Markov chain Monte Carlo, but these can be expensive when the data set is large and/or the model is complex, so more efficient variational approximations have recently received considerable attention. The traditional variational methods, that seek to minimize the Kullback--Leibler divergence between the posterior and a relatively simple parametric family, provide accurate and efficient estimation of the posterior mean, but often does not capture other moments, and have limitations in terms of the models to which they can be applied. Here we propose the construction of variational approximations based on minimizing the Fisher divergence, and develop an efficient computational algorithm that can be applied to a wide range of models without conjugacy or potentially unrealistic mean-field assumptions. We demonstrate the superior performance of the proposed method for the benchmark case of logistic regression.

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arXiv2019

Data-driven priors and their posterior concentration rates

In high-dimensional problems, choosing a prior distribution such that the corresponding posterior has desirable practical and theoretical properties can be challenging. This begs the question: can the data be used to help choose a good prior? In this paper, we develop a general strategy for constructing a data-driven or empirical prior and sufficient conditions for the corresponding posterior distribution to achieve a certain concentration rate. The idea is that the prior should put sufficient mass on parameter values for which the likelihood is large. An interesting byproduct of this data-driven centering is that the asymptotic properties of the posterior are less sensitive to the prior shape which, in turn, allows users to work with priors of computationally convenient forms while maintaining the desired rates. General results on both adaptive and non-adaptive rates based on empirical priors are presented, along with illustrations in density estimation, nonparametric regression, and high-dimensional structured normal models.

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arXiv2019

On optimal designs for non-regular models

Classically, Fisher information is the relevant object in defining optimal experimental designs. However, for models that lack certain regularity, the Fisher information does not exist and, hence, there is no notion of design optimality available in the literature. This article seeks to fill the gap by proposing a so-called Hellinger information, which generalizes Fisher information in the sense that the two measures agree in regular problems, but the former also exists for certain types of non-regular problems. We derive a Hellinger information inequality, showing that Hellinger information defines a lower bound on the local minimax risk of estimators. This provides a connection between features of the underlying model---in particular, the design---and the performance of estimators, motivating the use of this new Hellinger information for non-regular optimal design problems. Hellinger optimal designs are derived for several non-regular regression problems, with numerical results empirically demonstrating the efficiency of these designs compared to alternatives.

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arXiv2018

On nonparametric estimation of a mixing density via the predictive recursion algorithm

Nonparametric estimation of a mixing density based on observations from the corresponding mixture is a challenging statistical problem. This paper surveys the literature on a fast, recursive estimator based on the predictive recursion algorithm. After introducing the algorithm and giving a few examples, I summarize the available asymptotic convergence theory, describe an important semiparametric extension, and highlight two interesting applications. I conclude with a discussion of several recent developments in this area and some open problems.

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arXiv2018

Empirical Bayes posterior concentration in sparse high-dimensional linear models

We propose a new empirical Bayes approach for inference in the $p \gg n$ normal linear model. The novelty is the use of data in the prior in two ways, for centering and regularization. Under suitable sparsity assumptions, we establish a variety of concentration rate results for the empirical Bayes posterior distribution, relevant for both estimation and model selection. Computation is straightforward and fast, and simulation results demonstrate the strong finite-sample performance of the empirical Bayes model selection procedure.

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arXiv2018

Bayesian inference in high-dimensional linear models using an empirical correlation-adaptive prior

In the context of a high-dimensional linear regression model, we propose the use of an empirical correlation-adaptive prior that makes use of information in the observed predictor variable matrix to adaptively address high collinearity, determining if parameters associated with correlated predictors should be shrunk together or kept apart. Under suitable conditions, we prove that this empirical Bayes posterior concentrates around the true sparse parameter at the optimal rate asymptotically. A simplified version of a shotgun stochastic search algorithm is employed to implement the variable selection procedure, and we show, via simulation experiments across different settings and a real-data application, the favorable performance of the proposed method compared to existing methods.

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arXiv2018

Empirical priors and posterior concentration rates for a monotone density

In a Bayesian context, prior specification for inference on monotone densities is conceptually straightforward, but proving posterior convergence theorems is complicated by the fact that desirable prior concentration properties often are not satisfied. In this paper, I first develop a new prior designed specifically to satisfy an empirical version of the prior concentration property, and then I give sufficient conditions on the prior inputs such that the corresponding empirical Bayes posterior concentrates around the true monotone density at nearly the optimal minimax rate. Numerical illustrations also reveal the practical benefits of the proposed empirical Bayes approach compared to Dirichlet process mixtures.

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arXiv2018

Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image

Detection of an image boundary when the pixel intensities are measured with noise is an important problem in image segmentation, with numerous applications in medical imaging and engineering. From a statistical point of view, the challenge is that likelihood-based methods require modeling the pixel intensities inside and outside the image boundary, even though these are typically of no practical interest. Since misspecification of the pixel intensity models can negatively affect inference on the image boundary, it would be desirable to avoid this modeling step altogether. Towards this, we develop a robust Gibbs approach that constructs a posterior distribution for the image boundary directly, without modeling the pixel intensities. We prove that, for a suitable prior on the image boundary, the Gibbs posterior concentrates asymptotically at the minimax optimal rate, adaptive to the boundary smoothness. Monte Carlo computation of the Gibbs posterior is straightforward, and simulation experiments show that the corresponding inference is more accurate than that based on existing Bayesian methodology.

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arXiv2018

Calibrating general posterior credible regions

An advantage of methods that base inference on a posterior distribution is that credible regions are readily obtained. Except in well-specified situations, however, there is no guarantee that such regions will achieve the nominal frequentist coverage probability, even approximately. To overcome this difficulty, we propose a general strategy that introduces an additional scalar tuning parameter to control the posterior spread, and we develop an algorithm that chooses this parameter so that the corresponding credible region achieves the nominal coverage probability.

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arXiv2017

Fast nonparametric near-maximum likelihood estimation of a mixing density

Mixture models are regularly used in density estimation applications, but the problem of estimating the mixing distribution remains a challenge. Nonparametric maximum likelihood produce estimates of the mixing distribution that are discrete, and these may be hard to interpret when the true mixing distribution is believed to have a smooth density. In this paper, we investigate an algorithm that produces a sequence of smooth estimates that has been conjectured to converge to the nonparametric maximum likelihood estimator. Here we give a rigorous proof of this conjecture, and propose a new data-driven stopping rule that produces smooth near-maximum likelihood estimates of the mixing density, and simulations demonstrate the quality empirical performance of this estimator.

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arXiv2017

On overfitting and post-selection uncertainty assessments

In a regression context, when the relevant subset of explanatory variables is uncertain, it is common to use a data-driven model selection procedure. Classical linear model theory, applied naively to the selected sub-model, may not be valid because it ignores the selected sub-model's dependence on the data. We provide an explanation of this phenomenon, in terms of overfitting, for a class of model selection criteria.

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