  Brute force cross-validation (CV) is a method for predictive assessment and
model selection that is general and applicable to a wide range of Bayesian
models. Naive or `brute force' CV approaches are often too computationally
costly for interactive modeling workflows, especially when inference relies on
Markov chain Monte Carlo (MCMC). We propose overcoming this limitation using
massively parallel MCMC. Using accelerator hardware such as graphics processor
units (GPUs), our approach can be about as fast (in wall clock time) as a
single full-data model fit.
  Parallel CV is flexible because it can easily exploit a wide range data
partitioning schemes, such as those designed for non-exchangeable data. It can
also accommodate a range of scoring rules.
  We propose MCMC diagnostics, including a summary of MCMC mixing based on the
popular potential scale reduction factor ($\widehat{R}$) and MCMC effective
sample size ($\widehat{ESS}$) measures. We also describe a method for
determining whether an $\widehat{R}$ diagnostic indicates approximate
stationarity of the chains, that may be of more general interest for
applications beyond parallel CV. Finally, we show that parallel CV and its
diagnostics can be implemented with online algorithms, allowing parallel CV to
scale up to very large blocking designs on memory-constrained computing
accelerators.
