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We analyze the complexity of Gibbs samplers for inference in crossed randomeffect models used in modern analysis of variance. We demonstrate that forcertain designs the plain vanilla Gibbs sampler is not scalable, in the sensethat its complexity is worse than proportional to the number of parameters anddata. We thus propose a simple modification leading to a collapsed Gibbssampler that is provably scalable. Although our theory requires somebalancedness assumptions on the data designs, we demonstrate in simulated andreal datasets that the rates it predicts match remarkably the correct rates incases where the assumptions are violated. We also show that the collapsed Gibbssampler, extended to sample further unknown hyperparameters, outperformssignificantly alternative state of the art algorithms.

Scalable inference for crossed random effects models