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We consider exact algorithms for Bayesian inference with model selectionpriors (including spike-and-slab priors) in the sparse normal sequence model.Because the best existing exact algorithm becomes numerically unstable forsample sizes over n=500, there has been much attention for alternativeapproaches like approximate algorithms (Gibbs sampling, variational Bayes,etc.), shrinkage priors (e.g. the Horseshoe prior and the Spike-and-Slab LASSO)or empirical Bayesian methods. However, by introducing algorithmic ideas fromonline sequential prediction, we show that exact calculations are feasible formuch larger sample sizes: for general model selection priors we reach n=25000,and for certain spike-and-slab priors we can easily reach n=1. We furtherprove a de Finetti-like result for finite sample sizes that characterizesexactly which model selection priors can be expressed as spike-and-slab priors.The computational speed and numerical accuracy of the proposed methods aredemonstrated in experiments on simulated data, on a differential geneexpression data set, and to compare the effect of multiple hyper-parametersettings in the beta-binomial prior. In our experimental evaluation we computeguaranteed bounds on the numerical accuracy of all new algorithms, which showsthat the proposed methods are numerically reliable whereas an alternative basedon long division is not.

Fast Exact Bayesian Inference for Sparse Signals in the Normal Sequence Model