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We describe a class of algorithms for evaluating posterior moments of certainBayesian linear regression models with a normal likelihood and a normal prioron the regression coefficients. The proposed methods can be used forhierarchical mixed effects models with partial pooling over one group ofpredictors, as well as random effects models with partial pooling over twogroups of predictors. We demonstrate the performance of the methods on twoapplications, one involving U.S. opinion polls and one involving the modelingof COVID-19 outbreaks in Israel using survey data. The algorithms involveanalytical marginalization of regression coefficients followed by numericalintegration of the remaining low-dimensional density. The dominant cost of thealgorithms is an eigendecomposition computed once for each value of the outsideparameter of integration. Our approach drastically reduces run times comparedto state-of-the-art Markov chain Monte Carlo (MCMC) algorithms. The latter, inaddition to being computationally expensive, can also be difficult to tune whenapplied to hierarchical models.

Fast methods for posterior inference of two-group normal-normal models