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

TL;DR

This paper introduces a novel algorithmic approach inspired by online sequential prediction that enables exact Bayesian inference for sparse signals in the normal sequence model at much larger sample sizes than previously possible, with demonstrated numerical reliability.

Contribution

The authors develop a new exact inference algorithm that significantly extends the feasible sample size for Bayesian model selection priors, outperforming existing methods in stability and accuracy.

Findings

Exact calculations feasible for n=25000 with general priors

Exact calculations feasible for n=100000 with certain spike-and-slab priors

Demonstrated numerical reliability and stability of the proposed algorithms

Abstract

We consider exact algorithms for Bayesian inference with model selection priors (including spike-and-slab priors) in the sparse normal sequence model. Because the best existing exact algorithm becomes numerically unstable for sample sizes over n=500, there has been much attention for alternative approaches 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 from online sequential prediction, we show that exact calculations are feasible for much larger sample sizes: for general model selection priors we reach n=25000, and for certain spike-and-slab priors we can easily reach n=100000. We further prove a de Finetti-like result for finite sample sizes that characterizes exactly which model selection priors can be expressed as spike-and-slab priors. The computational speed and numerical accuracy of the proposed methods are demonstrated in experiments on simulated data, on a differential gene expression data set, and to compare the effect of multiple hyper-parameter settings in the beta-binomial prior. In our experimental evaluation we compute guaranteed bounds on the numerical accuracy of all new algorithms, which shows that the proposed methods are numerically reliable whereas an alternative based on long division is not.