arXiv subjects

Matthew Stephens

Publications and source records attributed to Matthew Stephens.

At least 19 recordsLinked to original sources

arXiv2026

Empirical Bayes Shrinkage of Functional Effects, with Application to Analysis of Dynamic eQTLs

We introduce functional adaptive shrinkage (FASH), an empirical Bayes method for joint analysis of observation units in which each unit estimates an effect function at several values of a continuous condition variable. The ideas in this paper are motivated by dynamic expression quantitative trait locus (eQTL) studies, which aim to characterize how genetic effects on gene expression vary with time or another continuous condition. FASH integrates a broad family of Gaussian processes defined through linear differential operators into an empirical Bayes shrinkage framework, enabling adaptive smoothing and borrowing of information across units. This provides improved estimation of effect functions and principled hypothesis testing, allowing straightforward computation of significance measures such as local false discovery and false sign rates. To encourage conservative inferences, we propose a simple prior- adjustment method that has theoretical guarantees and can be more broadly used with other empirical Bayes methods. We illustrate the benefits of FASH by reanalyzing dynamic eQTL data on cardiomyocyte differentiation from induced pluripotent stem cells. FASH identified novel dynamic eQTLs, revealed diverse temporal effect patterns, and provided improved power compared with the original analysis. More broadly, FASH offers a flexible statistical framework for joint analysis of functional data, with applications extending beyond genomics. To facilitate use of FASH in dynamic eQTL studies and other settings, we provide an accompanying R package at https: //github.com/stephenslab/fashr.

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arXiv2026

A Flexible Empirical Bayes Approach to Generalized Linear Models, with Applications to Sparse Logistic Regression

We introduce a flexible empirical Bayes approach for fitting Bayesian generalized linear models. Specifically, we adopt a novel mean-field variational inference (VI) method and the prior is estimated within the VI algorithm, making the method tuning-free. Unlike traditional VI methods that optimize the posterior density function, our approach directly optimizes the posterior mean and prior parameters. This formulation reduces the number of parameters to optimize and enables the use of scalable algorithms such as L-BFGS and stochastic gradient descent. Furthermore, our method automatically determines the optimal posterior based on the prior and likelihood, distinguishing it from existing VI methods that often assume a Gaussian variational. Our approach represents a unified framework applicable to a wide range of exponential family distributions, removing the need to develop unique VI methods for each combination of likelihood and prior distributions. We apply the framework to solve sparse logistic regression and demonstrate the superior predictive performance of our method in extensive numerical studies, by comparing it to prevalent sparse logistic regression approaches.

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arXiv2026

A New Family of Poisson Non-negative Matrix Factorization Methods Using the Shifted Log Link

Poisson non-negative matrix factorization (NMF) is a widely used method to find interpretable "parts-based" decompositions of count data. While many variants of Poisson NMF exist, existing methods assume that the "parts" in the decomposition combine additively. This assumption may be natural in some settings, but not in others. Here we introduce Poisson NMF with the shifted-log link function to relax this assumption. The shifted-log link function has a single tuning parameter, and as this parameter varies the model changes from assuming that parts combine additively (i.e., standard Poisson NMF) to assuming that parts combine more multiplicatively. We provide an algorithm to fit this model by maximum likelihood, and also an approximation that substantially reduces computation time for large, sparse datasets (computations scale with the number of non-zero entries in the data matrix). We illustrate these new methods on a variety of real datasets. Our examples show how the choice of link function in Poisson NMF can substantively impact the results, and how in some settings the use of a shifted-log link function may improve interpretability compared with the standard, additive link.

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arXiv2025

Bayesian variable selection in a Cox proportional hazards model with the "Sum of Single Effects" prior

Motivated by genetic fine-mapping applications, we introduce a new approach to Bayesian variable selection regression (BVSR) for time-to-event (TTE) outcomes. This new approach is designed to deal with the specific challenges that arise in genetic fine-mapping, including: the presence of very strong correlations among the covariates, often exceeding 0.99; very large data sets containing potentially thousands of covariates and hundreds of thousands of samples. We accomplish this by extending the "Sum of Single Effects" (SuSiE) method to the Cox proportional hazards (CoxPH) model. We demonstrate the benefits of the new method, "CoxPH-SuSiE", over existing BVSR methods for TTE outcomes in simulated fine-mapping data sets. We also illustrate CoxPH-SuSiE on real data by fine-mapping asthma loci using data from UK Biobank. This fine-mapping identified 14 asthma risk SNPs in 8 asthma risk loci, among which 6 had strong evidence for being causal (posterior inclusion probability greater than 50%). Two of the 6 putatively causal variants are known to be pathogenic, and others lie within a genomic sequence that is known to regulate the expression of GATA3.

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arXiv2025

Covariate-moderated Empirical Bayes Matrix Factorization

Matrix factorization is a fundamental method in statistics and machine learning for inferring and summarizing structure in multivariate data. Modern data sets often come with "side information" of various forms (images, text, graphs) that can be leveraged to improve estimation of the underlying structure. However, existing methods that leverage side information are limited in the types of data they can incorporate, and they assume specific parametric models. Here, we introduce a novel method for this problem, covariate-moderated empirical Bayes matrix factorization (cEBMF). cEBMF is a modular framework that accepts any type of side information that is processable by a probabilistic model or a neural network. The cEBMF framework can accommodate different assumptions and constraints on the factors through the use of different priors, and it adapts these priors to the data. We demonstrate the benefits of cEBMF in simulations and in analyses of spatial transcriptomics and collaborative filtering data. A PyTorch-based implementation of cEBMF with flexible priors is available at https://github.com/william-denault/cebmf_torch.

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arXiv2024

Gradient-based optimization for variational empirical Bayes multiple regression

Variational empirical Bayes (VEB) methods provide a practically attractive approach to fitting large, sparse, multiple regression models. These methods usually use coordinate ascent to optimize the variational objective function, an approach known as coordinate ascent variational inference (CAVI). Here we propose alternative optimization approaches based on gradient-based (quasi-Newton) methods, which we call gradient-based variational inference (GradVI). GradVI exploits a recent result from Kim et. al. [arXiv:2208.10910] which writes the VEB regression objective function as a penalized regression. Unfortunately the penalty function is not available in closed form, and we present and compare two approaches to dealing with this problem. In simple situations where CAVI performs well, we show that GradVI produces similar predictive performance, and GradVI converges in fewer iterations when the predictors are highly correlated. Furthermore, unlike CAVI, the key computations in GradVI are simple matrix-vector products, and so GradVI is much faster than CAVI in settings where the design matrix admits fast matrix-vector products (e.g., as we show here, trendfiltering applications) and lends itself to parallelized implementations in ways that CAVI does not. GradVI is also very flexible, and could exploit automatic differentiation to easily implement different prior families. Our methods are implemented in an open-source Python software, GradVI (available from https://github.com/stephenslab/gradvi ).

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arXiv2024

Improved methods for empirical Bayes multivariate multiple testing and effect size estimation

Estimating the sharing of genetic effects across different conditions is important to many statistical analyses of genomic data. The patterns of sharing arising from these data are often highly heterogeneous. To flexibly model these heterogeneous sharing patterns, Urbut et al. (2019) proposed the multivariate adaptive shrinkage (MASH) method to jointly analyze genetic effects across multiple conditions. However, multivariate analyses using MASH (as well as other multivariate analyses) require good estimates of the sharing patterns, and estimating these patterns efficiently and accurately remains challenging. Here we describe new empirical Bayes methods that provide improvements in speed and accuracy over existing methods. The two key ideas are: (1) adaptive regularization to improve accuracy in settings with many conditions; (2) improving the speed of the model fitting algorithms by exploiting analytical results on covariance estimation. In simulations, we show that the new methods provide better model fits, better out-of-sample performance, and improved power and accuracy in detecting the true underlying signals. In an analysis of eQTLs in 49 human tissues, our new analysis pipeline achieves better model fits and better out-of-sample performance than the existing MASH analysis pipeline. We have implemented the new methods, which we call ``Ultimate Deconvolution'', in an R package, udr, available on GitHub.

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arXiv2023

Empirical Bayes Covariance Decomposition, and a Solution to the Multiple Tuning Problem in Sparse PCA

Sparse Principal Components Analysis (PCA) has been proposed as a way to improve both interpretability and reliability of PCA. However, use of sparse PCA in practice is hindered by the difficulty of tuning the multiple hyperparameters that control the sparsity of different PCs (the "multiple tuning problem", MTP). Here we present a solution to the MTP using Empirical Bayes methods. We first introduce a general formulation for penalized PCA of a data matrix $\mathbf{X}$, which includes some existing sparse PCA methods as special cases. We show that this formulation also leads to a penalized decomposition of the covariance (or Gram) matrix, $\mathbf{X}^T\mathbf{X}$. We introduce empirical Bayes versions of these penalized problems, in which the penalties are determined by prior distributions that are estimated from the data by maximum likelihood rather than cross-validation. The resulting "Empirical Bayes Covariance Decomposition" provides a principled and efficient solution to the MTP in sparse PCA, and one that can be immediately extended to incorporate other structural assumptions (e.g. non-negative PCA). We illustrate the effectiveness of this approach on both simulated and real data examples.

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arXiv2022

A flexible model for correlated count data, with application to multi-condition differential expression analyses of single-cell RNA sequencing data

Detecting differences in gene expression is an important part of single-cell RNA sequencing experiments, and many statistical methods have been developed for this aim. Most differential expression analyses focus on comparing expression between two groups (e.g., treatment vs. control). But there is increasing interest in multi-condition differential expression analyses in which expression is measured in many conditions, and the aim is to accurately detect and estimate expression differences in all conditions. We show that directly modeling single-cell RNA-seq counts in all conditions simultaneously, while also inferring how expression differences are shared across conditions, leads to greatly improved performance for detecting and estimating expression differences compared to existing methods. We illustrate the potential of this new approach by analyzing data from a single-cell experiment studying the effects of cytokine stimulation on gene expression. We call our new method "Poisson multivariate adaptive shrinkage", and it is implemented in an R package available online at https://github.com/stephenslab/poisson.mash.alpha.

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arXiv2022

A flexible empirical Bayes approach to multiple linear regression and connections with penalized regression

We introduce a new empirical Bayes approach for large-scale multiple linear regression. Our approach combines two key ideas: (i) the use of flexible "adaptive shrinkage" priors, which approximate the nonparametric family of scale mixture of normal distributions by a finite mixture of normal distributions; and (ii) the use of variational approximations to efficiently estimate prior hyperparameters and compute approximate posteriors. Combining these two ideas results in fast and flexible methods, with computational speed comparable to fast penalized regression methods such as the Lasso, and with competitive prediction accuracy across a wide range of scenarios. Further, we provide new results that establish conceptual connections between our empirical Bayes methods and penalized methods. Specifically, we show that the posterior mean from our method solves a penalized regression problem, with the form of the penalty function being learned from the data by directly solving an optimization problem (rather than being tuned by cross-validation). Our methods are implemented in an R package, mr.ash.alpha, available from https://github.com/stephenslab/mr.ash.alpha.

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arXiv2021

ebnm: An R Package for Solving the Empirical Bayes Normal Means Problem Using a Variety of Prior Families

The empirical Bayes normal means (EBNM) model is important to many areas of statistics, including (but not limited to) multiple testing, wavelet denoising, and gene expression analysis. There are several existing software packages that can fit EBNM models under different prior assumptions and using different algorithms; however, the differences across interfaces complicate direct comparisons. Further, a number of important prior assumptions do not yet have implementations. Motivated by these issues, we developed the R package ebnm, which provides a unified interface for efficiently fitting EBNM models using a variety of prior assumptions, including nonparametric approaches. In some cases, we incorporated existing implementations into ebnm; in others, we implemented new fitting procedures with a focus on speed and numerical stability. We illustrate the use of ebnm in a detailed analysis of baseball statistics. By providing a unified and easily extensible interface, the ebnm package can facilitate development of new methods in statistics, genetics, and other areas; as an example, we briefly discuss the R package flashier, which harnesses methods in ebnm to provide a flexible and robust approach to matrix factorization.

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arXiv2021

Multi-scale Poisson process approaches for differential expression analysis of high-throughput sequencing data

Estimating and testing for differences in molecular phenotypes (e.g. gene expression, chromatin accessibility, transcription factor binding) across conditions is an important part of understanding the molecular basis of gene regulation. These phenotypes are commonly measured using high-throughput sequencing assays (e.g., RNA-seq, ATAC-seq, ChIP-seq), which provide high-resolution count data that reflect how the phenotypes vary along the genome. Multiple methods have been proposed to help exploit these high-resolution measurements for differential expression analysis. However, they ignore the count nature of the data, instead using normal approximations that work well only for data with large sample sizes or high counts. Here we develop count-based methods to address this problem. We model the data for each sample using an inhomogeneous Poisson process with spatially structured underlying intensity function, and then, building on multi-scale models for the Poisson process, estimate and test for differences in the underlying intensity function across samples (or groups of samples). Using both simulation and real ATAC-seq data we show that our method outperforms previous normal-based methods, especially in situations with small sample sizes or low counts.

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arXiv2021

Non-negative matrix factorization algorithms generally improve topic model fits

In an effort to develop topic modeling methods that can be quickly applied to large data sets, we revisit the problem of maximum-likelihood estimation in topic models. It is known, at least informally, that maximum-likelihood estimation in topic models is closely related to non-negative matrix factorization (NMF). Yet, to our knowledge, this relationship has not been exploited previously to fit topic models. We show that recent advances in NMF optimization methods can be leveraged to fit topic models very efficiently, often resulting in much better fits and in less time than existing algorithms for topic models. We also formally make the connection between the NMF optimization problem and maximum-likelihood estimation for the topic model, and using this result we show that the expectation maximization (EM) algorithm for the topic model is essentially the same as the classic multiplicative updates for NMF (the only difference being that the operations are performed in a different order). Our methods are implemented in the R package fastTopics.

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arXiv2019

Empirical Bayes estimation of normal means, accounting for uncertainty in estimated standard errors

We consider Empirical Bayes (EB) estimation in the normal means problem, when the standard deviations of the observations are not known precisely, but estimated with error -- which is almost always the case in practical applications. In classical statistics accounting for estimated standard errors usually involves replacing a normal distribution with a $t$ distribution. This suggests approaching this problem by replacing the normal assumption with a $t$ assumption, leading to an "EB $t$-means problem". Here we show that an approach along these lines can indeed work, but only with some care. Indeed, a naive application of this idea is flawed, and can perform poorly. We suggest how this flaw can be remedied by a two-stage procedure, which first performs EB shrinkage estimation of the standard errors and then solves an EB $t$-means problem. We give numerical results illustrating the effectiveness of this remedy.

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arXiv2018

Solving the Empirical Bayes Normal Means Problem with Correlated Noise

The Normal Means problem plays a fundamental role in many areas of modern high-dimensional statistics, both in theory and practice. And the Empirical Bayes (EB) approach to solving this problem has been shown to be highly effective, again both in theory and practice. However, almost all EB treatments of the Normal Means problem assume that the observations are independent. In practice correlations are ubiquitous in real-world applications, and these correlations can grossly distort EB estimates. Here, exploiting theory from Schwartzman (2010), we develop new EB methods for solving the Normal Means problem that take account of unknown correlations among observations. We provide practical software implementations of these methods, and illustrate them in the context of large-scale multiple testing problems and False Discovery Rate (FDR) control. In realistic numerical experiments our methods compare favorably with other commonly-used multiple testing methods.

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arXiv2018

A fast algorithm for maximum likelihood estimation of mixture proportions using sequential quadratic programming

Maximum likelihood estimation of mixture proportions has a long history, and continues to play an important role in modern statistics, including in development of nonparametric empirical Bayes methods. Maximum likelihood of mixture proportions has traditionally been solved using the expectation maximization (EM) algorithm, but recent work by Koenker & Mizera shows that modern convex optimization techniques -- in particular, interior point methods -- are substantially faster and more accurate than EM. Here, we develop a new solution based on sequential quadratic programming (SQP). It is substantially faster than the interior point method, and just as accurate. Our approach combines several ideas: first, it solves a reformulation of the original problem; second, it uses an SQP approach to make the best use of the expensive gradient and Hessian computations; third, the SQP iterations are implemented using an active set method to exploit the sparse nature of the quadratic subproblems; fourth, it uses accurate low-rank approximations for more efficient gradient and Hessian computations. We illustrate the benefits of our approach in experiments on synthetic data sets as well as a large genetic association data set. In large data sets (n = 1,000,000 observations, m = 1,000 mixture components), our implementation achieves at least 100-fold reduction in runtime compared with a state-of-the-art interior point solver. Our methods are implemented in Julia, and in an R package available on CRAN (see https://CRAN.R-project.org/package=mixsqp).

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arXiv2018

Empirical Bayes Matrix Factorization

Matrix factorization methods - including Factor analysis (FA), and Principal Components Analysis (PCA) - are widely used for inferring and summarizing structure in multivariate data. Many matrix factorization methods exist, corresponding to different assumptions on the elements of the underlying matrix factors. For example, many recent methods use a penalty or prior distribution to achieve sparse representations ("Sparse FA/PCA"). Here we introduce a general Empirical Bayes approach to matrix factorization (EBMF), whose key feature is that it uses the observed data to estimate prior distributions on matrix elements. We derive a correspondingly-general variational fitting algorithm, which reduces fitting EBMF to solving a simpler problem - the so-called "normal means" problem. We implement this general algorithm, but focus particular attention on the use of sparsity-inducing priors that are uni-modal at 0. This yields a sparse EBMF approach - essentially a version of sparse FA/PCA - that automatically adapts the amount of sparsity to the data. We demonstrate the benefits of our approach through both numerical comparisons with competing methods and through analysis of data from the GTEx (Genotype Tissue Expression) project on genetic associations across 44 human tissues. In numerical comparisons EBMF often provides more accurate inferences than other methods. In the GTEx data, EBMF identifies interpretable structure that concords with known relationships among human tissues. Software implementing our approach is available at https://github.com/stephenslab/flashr

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arXiv2017

Empirical Bayes Shrinkage and False Discovery Rate Estimation, Allowing For Unwanted Variation

We combine two important ideas in the analysis of large-scale genomics experiments (e.g. experiments that aim to identify genes that are differentially expressed between two conditions). The first is use of Empirical Bayes (EB) methods to handle the large number of potentially-sparse effects, and estimate false discovery rates and related quantities. The second is use of factor analysis methods to deal with sources of unwanted variation such as batch effects and unmeasured confounders. We describe a simple modular fitting procedure that combines key ideas from both these lines of research. This yields new, powerful EB methods for analyzing genomics experiments that account for both sparse effects and unwanted variation. In realistic simulations, these new methods provide significant gains in power and calibration over competing methods. In real data analysis we find that different methods, while often conceptually similar, can vary widely in their assessments of statistical significance. This highlights the need for care in both choice of methods and interpretation of results. All methods introduced in this paper are implemented in the R package vicar available at https://github.com/dcgerard/vicar .

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