EbayesThresh: R Programs for Empirical Bayes Thresholding
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Abstract
Suppose that a sequence of unknown parameters is observed sub ject 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 heavy-tailed density. Within the package, this can be either a Laplace (double exponential) density or else a mixture of normal distributions with tail behavior similar to the Cauchy distribution. The mixing weight, or sparsity parameter, is chosen automatically by marginal maximum likelihood. If estimation is carried out using the posterior median, this is a random thresholding procedure; the estimation can also be carried out using other thresholding rules with the same threshold, and the package provides the posterior mean, and hard and soft thresholding, as additional options.
This paper reviews the method, and gives details (far beyond those previously published) of the calculations needed for implementing the procedures. It explains and motivates both the general methodology, and the use of the EbayesThresh package, through simulated and real data examples. When estimating the wavelet transform of an unknown function, it is appropriate to apply the method level by level to the transform of the observed data. The package can carry out these calculations for wavelet transforms obtained using various packages in R and S-PLUS. Details, including a motivating example, are presented, and the application of the method to image estimation is also explored.
The final topic considered is the estimation of a single sequence that may become progressively sparser along the sequence. An iterated least squares isotone regression method allows for the choice of a threshold that depends monotonically on the order in which the observations are made. An alternative possibility, also discussed in detail, is a particular parametric dependence of the sparsity parameter on the position in the sequence.
The prior for each parameter in the sequence is a mixture of an atom of probability at zero and a heavy-tailed density. Within the package, this can be either a Laplace (double exponential) density or else a mixture of normal distributions with tail behavior similar to the Cauchy distribution. The mixing weight, or sparsity parameter, is chosen automatically by marginal maximum likelihood. If estimation is carried out using the posterior median, this is a random thresholding procedure; the estimation can also be carried out using other thresholding rules with the same threshold, and the package provides the posterior mean, and hard and soft thresholding, as additional options.
This paper reviews the method, and gives details (far beyond those previously published) of the calculations needed for implementing the procedures. It explains and motivates both the general methodology, and the use of the EbayesThresh package, through simulated and real data examples. When estimating the wavelet transform of an unknown function, it is appropriate to apply the method level by level to the transform of the observed data. The package can carry out these calculations for wavelet transforms obtained using various packages in R and S-PLUS. Details, including a motivating example, are presented, and the application of the method to image estimation is also explored.
The final topic considered is the estimation of a single sequence that may become progressively sparser along the sequence. An iterated least squares isotone regression method allows for the choice of a threshold that depends monotonically on the order in which the observations are made. An alternative possibility, also discussed in detail, is a particular parametric dependence of the sparsity parameter on the position in the sequence.