Backfitting for large scale crossed random effects regressions

Regression models with crossed random effect errors can be very expensive to compute. The cost of both generalized least squares and Gibbs sampling easily grow as N3/2 (or worse) for N observations. Papaspiliopoulos, Roberts Zanella (Biometrika 107 (2020) 25–40) present a collapsed sampler that costs O(N), but under an extremely stringent model. We propose backfitting algorithm compute estimate prove it O(N). A critical part the proof is in ensuring number iterations required O(1), which follows from keeping certain matrix norm below 1?? some ?>0. Our conditions are greatly relaxed compared those sampler, though still strict. Empirically, has less strict than our assumptions. illustrate new on ratings data set Stitch Fix.