In recent years, a rich variety of shrinkage priors have been proposed that have great promise in addressing massive regression problems. In general, these new priors can be expressed as scale mixtures of normals, but have more complex forms and better properties than traditional Cauchy and double exponential priors. We first propose a new class of normal scale mixtures through a novel generalized beta distribution that encompasses many interesting priors as special cases. This encompassing framework should prove useful in comparing competing priors, considering properties and revealing close connections. We then develop a class of variational Bayes approximations through the new hierarchy presented that will scale more efficiently to the types of truly massive data sets that are now encountered routinely.

BACKGROUND: Tag-based techniques, such as SAGE, are commonly used to sample the mRNA pool of an organism's transcriptome. Incomplete digestion during the tag formation process may allow for multiple tags to be generated from a given mRNA transcript. The probability of forming a tag varies with its relative location. As a result, the observed tag counts represent a biased sample of the actual transcript pool. In SAGE this bias can be avoided by ignoring all but the 3' most tag but will discard a large fraction of the observed data. Taking this bias into account should allow more of the available data to be used leading to increased statistical power. RESULTS: Three new hierarchical models, which directly embed a model for the variation in tag formation probability, are proposed and their associated Bayesian inference algorithms are developed. These models may be applied to libraries at both the tag and aggregate level. Simulation experiments and analysis of real data are used to contrast the accuracy of the various methods. The consequences of tag formation bias are discussed in the context of testing differential expression. A description is given as to how these algorithms can be applied in that context. CONCLUSIONS: Several Bayesian inference algorithms that account for tag formation effects are compared with the DPB algorithm providing clear evidence of superior performance. The accuracy of inferences when using a particular non-informative prior is found to depend on the expression level of a given gene. The multivariate nature of the approach easily allows both univariate and joint tests of differential expression. Calculations demonstrate the potential for false positive and negative findings due to variation in tag formation probabilities across samples when testing for differential expression.