arXiv subjects

David Gerard

Publications and source records attributed to David Gerard.

6 recordsLinked to original sources

arXiv2026

Constructing and extending $n$ = 1 Bayesian confidence intervals for location parameters in location-scale families

It is a surprising, modestly known fact that when given a single observation from a normal distribution with unknown mean and unknown variance, valid and non-trivial confidence intervals for the mean can be constructed. These intervals are presented in papers fully formed, providing limited intuition for how they arise or how to generalize them. We show that these intervals can be constructed in a principled way using two separate Bayesian reasonings. In the first, for any continuous symmetric location-scale family (under mild regularity conditions) with $n=1$ observation, we derive priors which produce $(1 - \alpha)100\%$ credible intervals that are, asymptotically in the confidence level $\alpha \rightarrow 0$, valid $(1 - \alpha)100\%$ confidence intervals. In the second, we show that the $n=1$ frequentist intervals can be seen as $t$-intervals augmented with a prior value, and that these augmented $t$-intervals are equivalent to inverted frequentist tests using a Bayes factor (using appropriate priors) as a test statistic. For $n \geq 2$, our credible interval approach does not maintain the confidence level. However, for $n \geq 2$, our augmented $t$-intervals produce valid confidence intervals with lower expected squared width in parts of the parameter space than the Student $t$-intervals, indicating improvements when prior knowledge is available. We demonstrate these methods on an $n = 3$ dataset of hyperbolic excess velocities of interstellar objects.

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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.

stat.ME↗

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 .

stat.ME↗

arXiv2017

Unifying and Generalizing Methods for Removing Unwanted Variation Based on Negative Controls

Unwanted variation, including hidden confounding, is a well-known problem in many fields, particularly large-scale gene expression studies. Recent proposals to use control genes --- genes assumed to be unassociated with the covariates of interest --- have led to new methods to deal with this problem. Going by the moniker Removing Unwanted Variation (RUV), there are many versions --- RUV1, RUV2, RUV4, RUVinv, RUVrinv, RUVfun. In this paper, we introduce a general framework, RUV*, that both unites and generalizes these approaches. This unifying framework helps clarify connections between existing methods. In particular we provide conditions under which RUV2 and RUV4 are equivalent. The RUV* framework also preserves an advantage of RUV approaches --- their modularity --- which facilitates the development of novel methods based on existing matrix imputation algorithms. We illustrate this by implementing RUVB, a version of RUV* based on Bayesian factor analysis. In realistic simulations based on real data we found that RUVB is competitive with existing methods in terms of both power and calibration, although we also highlight the challenges of providing consistently reliable calibration among data sets.

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arXiv2015

Adaptive Higher-order Spectral Estimators

Many applications involve estimation of a signal matrix from a noisy data matrix. In such cases, it has been observed that estimators that shrink or truncate the singular values of the data matrix perform well when the signal matrix has approximately low rank. In this article, we generalize this approach to the estimation of a tensor of parameters from noisy tensor data. We develop new classes of estimators that shrink or threshold the mode-specific singular values from the higher-order singular value decomposition. These classes of estimators are indexed by tuning parameters, which we adaptively choose from the data by minimizing Stein's unbiased risk estimate. In particular, this procedure provides a way to estimate the multilinear rank of the underlying signal tensor. Using simulation studies under a variety of conditions, we show that our estimators perform well when the mean tensor has approximately low multilinear rank, and perform competitively when the signal tensor does not have approximately low multilinear rank. We illustrate the use of these methods in an application to multivariate relational data.

stat.ME↗

arXiv2014

Equivariant minimax dominators of the MLE in the array normal model

Inference about dependencies in a multiway data array can be made using the array normal model, which corresponds to the class of multivariate normal distributions with separable covariance matrices. Maximum likelihood and Bayesian methods for inference in the array normal model have appeared in the literature, but there have not been any results concerning the optimality properties of such estimators. In this article, we obtain results for the array normal model that are analogous to some classical results concerning covariance estimation for the multivariate normal model. We show that under a lower triangular product group, a uniformly minimum risk equivariant estimator (UMREE) can be obtained via a generalized Bayes procedure. Although this UMREE is minimax and dominates the MLE, it can be improved upon via an orthogonally equivariant modification. Numerical comparisons of the risks of these estimators show that the equivariant estimators can have substantially lower risks than the MLE.

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