Robbins’s visionary 1951 paper can be seen as an exercise in binary classification, but also as a precursor to the outpouring of recent work on high-dimensional data analysis and multiple testing. It can also be seen as the birth of empirical Bayes methods. Our objective in the present note is to use this problem and several variants of it to provide a glimpse into the evolution of empirical Ba...

Let X be a random variable which for simplicity we shall assume to have discrete values x and which has a probability distribution depending in a known way on an unknown real parameter A, (1) p (xIX) =Pr [X = xIA =X], A-itself being a random variable with a priori distribution function (2) G (X) =Pr [A-< X. The unconditional probability distribution of X is then given by (3) PG(x) =Pr[X=xI =fP(...

Suppose that a sequence of unknown parameters is observed subject to independent Gaussian noise. The EbayesThresh package in the S language implements a class of Empirical Bayes thresholding methods that can take advantage of possible sparsity in the sequence, to improve the quality of estimation. The prior for each parameter in the sequence is a mixture of an atom of probability at zero and a ...

BACKGROUND Advances in mass spectrometry-based proteomics have enabled the incorporation of proteomic data into systems approaches to biology. However, development of analytical methods has lagged behind. Here we describe an empirical Bayes framework for quantitative proteomics data analysis. The method provides a statistical description of each experiment, including the number of proteins that...

Consider the problem of estimating normal means from independent observations with known variances, possibly different from each other. Suppose that a second-level normal model is specified on the unknown means, with the prior means depending on a vector of covariates and the prior variances constant. For this two-level normal model, existing empirical Bayes methods are constructed from the Bay...

Bayesian methods based on hierarchical mixture models have demonstrated excellent mean squared error properties in constructing data dependent shrinkage estimators in wavelets, however, subjective elicitation of the hyperparameters is challenging. In this chapter we use an Empirical Bayes approach to estimate the hyperparameters for each level of the wavelet decomposition, bypassing the usual d...

In recent microarray experiments thousands of gene expressions are simultaneously tested in comparing samples (e.g., tissue types or experimental conditions). Application of a statistical test, such as the t-test, would lead to a p-value for each gene that reflects the amount of statistical evidence present in the data that the given gene is indeed differentially expressed. We show how to use t...

Empirical Bayes methods use the data from parallel experiments, for instance observations Xk ~ 𝒩 (Θ k , 1) for k = 1, 2, …, N, to estimate the conditional distributions Θ k |Xk . There are two main estimation strategies: modeling on the θ space, called "g-modeling" here, and modeling on the×space, called "f-modeling." The two approaches are de- scribed and compared. A series of computational fo...

This paper explores a class of empirical Bayes methods for leveldependent threshold selection in wavelet shrinkage. The prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy-tailed density. The mixing weight, or sparsity parameter, for each level of the transform is chosen by marginal maximum likelihood. If estimation is carried out using the p...

The domain of link analysis has recently re-ignited interest among researchers due to its applicability to new areas such as intelligence analysis (for example, identifying cliques of suspicious people), large scale social network analysis and genomics. The area of link analysis is not new and comprise a number of techniques developed by different communities. In this paper we propose a statist...