arXiv subjects

Steven N. MacEachern

Publications and source records attributed to Steven N. MacEachern.

At least 19 recordsLinked to original sources

arXiv2024

Bayesian Model Calibration and Sensitivity Analysis for Oscillating Biological Experiments

Understanding the oscillating behaviors that govern organisms' internal biological processes requires interdisciplinary efforts combining both biological and computer experiments, as the latter can complement the former by simulating perturbed conditions with higher resolution. Harmonizing the two types of experiment, however, poses significant statistical challenges due to identifiability issues, numerical instability, and ill behavior in high dimension. This article devises a new Bayesian calibration framework for oscillating biochemical models. The proposed Bayesian model is estimated relying on an advanced Markov chain Monte Carlo (MCMC) technique which can efficiently infer the parameter values that match the simulated and observed oscillatory processes. Also proposed is an approach to sensitivity analysis based on the intervention posterior. This approach measures the influence of individual parameters on the target process by using the obtained MCMC samples as a computational tool. The proposed framework is illustrated with circadian oscillations observed in a filamentous fungus, Neurospora crassa.

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arXiv2024

Familial inference: tests for hypotheses on a family of centres

Statistical hypotheses are translations of scientific hypotheses into statements about one or more distributions, often concerning their centre. Tests that assess statistical hypotheses of centre implicitly assume a specific centre, e.g., the mean or median. Yet, scientific hypotheses do not always specify a particular centre. This ambiguity leaves the possibility for a gap between scientific theory and statistical practice that can lead to rejection of a true null. In the face of replicability crises in many scientific disciplines, significant results of this kind are concerning. Rather than testing a single centre, this paper proposes testing a family of plausible centres, such as that induced by the Huber loss function (the Huber family). Each centre in the family generates a testing problem, and the resulting family of hypotheses constitutes a familial hypothesis. A Bayesian nonparametric procedure is devised to test familial hypotheses, enabled by a novel pathwise optimization routine to fit the Huber family. The favourable properties of the new test are demonstrated theoretically and experimentally. Two examples from psychology serve as real-world case studies.

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arXiv2023

Regularized Exponentially Tilted Empirical Likelihood for Bayesian Inference

Bayesian inference with empirical likelihood faces a challenge as the posterior domain is a proper subset of the original parameter space due to the convex hull constraint. We propose a regularized exponentially tilted empirical likelihood to address this issue. Our method removes the convex hull constraint using a novel regularization technique, incorporating a continuous exponential family distribution to satisfy a Kullback--Leibler divergence criterion. The regularization arises as a limiting procedure where pseudo-data are added to the formulation of exponentially tilted empirical likelihood in a structured fashion. We show that this regularized exponentially tilted empirical likelihood retains certain desirable asymptotic properties with improved finite sample performance. Simulation and data analysis demonstrate that the proposed method provides a suitable pseudo-likelihood for Bayesian inference.

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arXiv2023

A Process of Dependent Quantile Pyramids

Despite the practicality of quantile regression (QR), simultaneous estimation of multiple QR curves continues to be challenging. We address this problem by proposing a Bayesian nonparametric framework that generalizes the quantile pyramid by replacing each scalar variate in the quantile pyramid with a stochastic process on a covariate space. We propose a novel approach to show the existence of a quantile pyramid for all quantiles. The process of dependent quantile pyramids allows for non-linear QR and automatically ensures non-crossing of QR curves on the covariate space. Simulation studies document the performance and robustness of our approach. An application to cyclone intensity data is presented.

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arXiv2023

Empirical likelihood for the analysis of experimental designs

Empirical likelihood enables a nonparametric, likelihood-driven style of inference without restrictive assumptions routinely made in parametric models. We develop a framework for applying empirical likelihood to the analysis of experimental designs, addressing issues that arise from blocking and multiple hypothesis testing. In addition to popular designs such as balanced incomplete block designs, our approach allows for highly unbalanced, incomplete block designs. We derive an asymptotic multivariate chi-square distribution for a set of empirical likelihood test statistics and propose two single-step multiple testing procedures: asymptotic Monte Carlo and nonparametric bootstrap. Both procedures asymptotically control the generalised family-wise error rate and efficiently construct simultaneous confidence intervals for comparisons of interest without explicitly considering the underlying covariance structure. A simulation study demonstrates that the performance of the procedures is robust to violations of standard assumptions of linear mixed models. We also present an application to experiments on a pesticide.

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arXiv2022

Superset model problem

This paper focuses on the superset model problem that arises in the context of regression. To address this problem, we take the Bayesian approach to measure its uncertainty. An illustrative example with the real dataset is provided.

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arXiv2022

A regression approach to the two-dataset problem

This paper considers the two-dataset problem, where data are collected from two potentially different populations sharing common aspects. This problem arises when data are collected by two different types of researchers or from two different sources. We may reach invalid conclusions without using knowledge about the data collection process. To address this problem, this paper develops statistical regression models focusing on the difference in measurement and proposes two prediction errors that help to evaluate the underlying data collection process. As a consequence, it is possible to discuss the heterogeneity/similarity of the set of predictors in terms of prediction. Two real datasets are selected to illustrate our method.

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arXiv2022

Rediscovering a little known fact about the t-test and the F-test: Algebraic, Geometric, Distributional and Graphical Considerations

We discuss the role that the null hypothesis should play in the construction of a test statistic used to make a decision about that hypothesis. To construct the test statistic for a point null hypothesis about a binomial proportion, a common recommendation is to act as if the null hypothesis is true. We argue that, on the surface, the one-sample t-test of a point null hypothesis about a Gaussian population mean does not appear to follow the recommendation. We show how simple algebraic manipulations of the usual t-statistic lead to an equivalent test procedure consistent with the recommendation. We provide geometric intuition regarding this equivalence and we consider extensions to testing nested hypotheses in Gaussian linear models. We discuss an application to graphical residual diagnostics where the form of the test statistic makes a practical difference. By examining the formulation of the test statistic from multiple perspectives in this familiar example, we provide simple, concrete illustrations of some important issues that can guide the formulation of effective solutions to more complex statistical problems.

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arXiv2021

Economic variable selection

Regression plays a key role in many research areas and its variable selection is a classic and major problem. This study emphasizes cost of predictors to be purchased for future use, when we select a subset of them. Its economic aspect is naturally formalized by the decision-theoretic approach. In addition, two Bayesian approaches are proposed to address uncertainty about model parameters and models: the restricted and extended approaches, which lead us to rethink about model averaging. From objective, rule-based, or robust Bayes point of view, the former is preferred. Proposed method is applied to three popular datasets for illustration.

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arXiv2020

The Dependent Dirichlet Process and Related Models

Standard regression approaches assume that some finite number of the response distribution characteristics, such as location and scale, change as a (parametric or nonparametric) function of predictors. However, it is not always appropriate to assume a location/scale representation, where the error distribution has unchanging shape over the predictor space. In fact, it often happens in applied research that the distribution of responses under study changes with predictors in ways that cannot be reasonably represented by a finite dimensional functional form. This can seriously affect the answers to the scientific questions of interest, and therefore more general approaches are indeed needed. This gives rise to the study of fully nonparametric regression models. We review some of the main Bayesian approaches that have been employed to define probability models where the complete response distribution may vary flexibly with predictors. We focus on developments based on modifications of the Dirichlet process, historically termed dependent Dirichlet processes, and some of the extensions that have been proposed to tackle this general problem using nonparametric approaches.

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arXiv2019

Aggregated Pairwise Classification of Statistical Shapes

The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to early formulations of the problem - with an object to be classified described as a vector in a relatively low-dimensional Euclidean space. Statistical shape data have two main properties that suggest a need for a novel approach: (i) shapes are inherently infinite dimensional with strong dependence among the positions of nearby points, and (ii) shape space is not Euclidean, but is fundamentally curved. To accommodate these features of the data, we work with the square-root velocity function of the curves to provide a useful formal description of the shape, pass to tangent spaces of the manifold of shapes at different projection points which effectively separate shapes for pairwise classification in the training data, and use principal components within these tangent spaces to reduce dimensionality. We illustrate the impact of the projection point and choice of subspace on the misclassification rate with a novel method of combining pairwise classifiers.

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arXiv2018

Bayesian Restricted Likelihood Methods: Conditioning on Insufficient Statistics in Bayesian Regression

Bayesian methods have proven themselves to be successful across a wide range of scientific problems and have many well-documented advantages over competing methods. However, these methods run into difficulties for two major and prevalent classes of problems: handling data sets with outliers and dealing with model misspecification. We outline the drawbacks of previous solutions to both of these problems and propose a new method as an alternative. When working with the new method, the data is summarized through a set of insufficient statistics, targeting inferential quantities of interest, and the prior distribution is updated with the summary statistics rather than the complete data. By careful choice of conditioning statistics, we retain the main benefits of Bayesian methods while reducing the sensitivity of the analysis to features of the data not captured by the conditioning statistics. For reducing sensitivity to outliers, classical robust estimators (e.g., M-estimators) are natural choices for conditioning statistics. A major contribution of this work is the development of a data augmented Markov chain Monte Carlo (MCMC) algorithm for the linear model and a large class of summary statistics. We demonstrate the method on simulated and real data sets containing outliers and subject to model misspecification. Success is manifested in better predictive performance for data points of interest as compared to competing methods.

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arXiv2017

Marginally Interpretable Generalized Linear Mixed Models

Two popular approaches for relating correlated measurements of a non-Gaussian response variable to a set of predictors are to fit a marginal model using generalized estimating equations and to fit a generalized linear mixed model by introducing latent random variables. The first approach is effective for parameter estimation, but leaves one without a formal model for the data with which to assess quality of fit or make predictions for future observations. The second approach overcomes the deficiencies of the first, but leads to parameter estimates that must be interpreted conditional on the latent variables. Further complicating matters, obtaining marginal summaries from a generalized linear mixed model often requires evaluation of an analytically intractable integral or use of attenuation factors that are not exact. We define a class of marginally interpretable generalized linear mixed models that lead to parameter estimates with a marginal interpretation while maintaining the desirable statistical properties of a conditionally-specified model. We discuss the form of these models under various common link functions and also address computational issues associated with these models. For logistic mixed effects models, we introduce an accurate and efficient method for evaluating the logistic-normal integral.

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arXiv2016

Bandwidth Selection for Kernel Density Estimation with a Markov Chain Monte Carlo Sample

Markov chain Monte Carlo samplers produce dependent streams of variates drawn from the limiting distribution of the Markov chain. With this as motivation, we introduce novel univariate kernel density estimators which are appropriate for the stationary sequences of dependent variates. We modify the asymptotic mean integrated squared error criterion to account for dependence and find that the modified criterion suggests data-driven adjustments to standard bandwidth selection methods. Simulation studies show that our proposed methods find bandwidths close to the optimal value while standard methods lead to smaller bandwidths and hence to undersmoothed density estimates. Empirically, the proposed methods have considerably smaller integrated mean squared error than do standard methods.

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arXiv2015

Block Hyper-g Priors in Bayesian Regression

The development of prior distributions for Bayesian regression has traditionally been driven by the goal of achieving sensible model selection and parameter estimation. The formalization of properties that characterize good performance has led to the development and popularization of thick tailed mixtures of g priors such as the Zellner--Siow and hyper-g priors. The properties of a particular prior are typically illuminated under limits on the likelihood or the prior. In this paper we introduce a new, conditional information asymptotic that is motivated by the common data analysis setting where at least one regression coefficient is much larger than others. We analyze existing mixtures of g priors under this limit and reveal two new behaviors, Essentially Least Squares (ELS) estimation and the Conditional Lindley's Paradox (CLP), and argue that these behaviors are, in general, undesirable. As the driver behind both of these behaviors is the use of a single, latent scale parameter that is common to all coefficients, we propose a block hyper-g prior, defined by first partitioning the covariates into groups and then placing independent hyper-g priors on the corresponding blocks of coefficients. We provide conditions under which ELS and the CLP are avoided by the new class of priors, and provide consistency results under traditional sample size asymptotics.

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arXiv2012

Regularization of Case-Specific Parameters for Robustness and Efficiency

Regularization methods allow one to handle a variety of inferential problems where there are more covariates than cases. This allows one to consider a potentially enormous number of covariates for a problem. We exploit the power of these techniques, supersaturating models by augmenting the "natural" covariates in the problem with an additional indicator for each case in the data set. We attach a penalty term for these case-specific indicators which is designed to produce a desired effect. For regression methods with squared error loss, an $\ell_1$ penalty produces a regression which is robust to outliers and high leverage cases; for quantile regression methods, an $\ell_2$ penalty decreases the variance of the fit enough to overcome an increase in bias. The paradigm thus allows us to robustify procedures which lack robustness and to increase the efficiency of procedures which are robust. We provide a general framework for the inclusion of case-specific parameters in regularization problems, describing the impact on the effective loss for a variety of regression and classification problems. We outline a computational strategy by which existing software can be modified to solve the augmented regularization problem, providing conditions under which such modification will converge to the optimum solution. We illustrate the benefits of including case-specific parameters in the context of mean regression and quantile regression through analysis of NHANES and linguistic data sets.

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arXiv2012

Restricting exchangeable nonparametric distributions

Distributions over exchangeable matrices with infinitely many columns, such as the Indian buffet process, are useful in constructing nonparametric latent variable models. However, the distribution implied by such models over the number of features exhibited by each data point may be poorly- suited for many modeling tasks. In this paper, we propose a class of exchangeable nonparametric priors obtained by restricting the domain of existing models. Such models allow us to specify the distribution over the number of features per data point, and can achieve better performance on data sets where the number of features is not well-modeled by the original distribution.

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arXiv2011

Bayesian Synthesis: Combining subjective analyses, with an application to ozone data

Bayesian model averaging enables one to combine the disparate predictions of a number of models in a coherent fashion, leading to superior predictive performance. The improvement in performance arises from averaging models that make different predictions. In this work, we tap into perhaps the biggest driver of different predictions---different analysts---in order to gain the full benefits of model averaging. In a standard implementation of our method, several data analysts work independently on portions of a data set, eliciting separate models which are eventually updated and combined through a specific weighting method. We call this modeling procedure Bayesian Synthesis. The methodology helps to alleviate concerns about the sizable gap between the foundational underpinnings of the Bayesian paradigm and the practice of Bayesian statistics. In experimental work we show that human modeling has predictive performance superior to that of many automatic modeling techniques, including AIC, BIC, Smoothing Splines, CART, Bagged CART, Bayes CART, BMA and LARS, and only slightly inferior to that of BART. We also show that Bayesian Synthesis further improves predictive performance. Additionally, we examine the predictive performance of a simple average across analysts, which we dub Convex Synthesis, and find that it also produces an improvement.

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