Variational Bayes Inference for Spatio-temporal Dynamic Factor Models

We investigate fast variational approximations for spatio–temporal dynamic factor models, benchmarking them against a Markov Chain Monte Carlo (MCMC) baseline. We propose two Coordinate Ascent Variational Inference (CAVI) schemes: (i) a mean-field variational Bayes (MFVB) that factorizes states and spatial loadings, and (ii) a structured variational Bayes (SVB) that preserves autoregressive state dependence while maintaining a mean-field structure across space. Using monthly NCEP/NCAR 1000 mb air temperature data over North America (1948–2025; 928 months; 204 locations), we find that SVB closely reproduces the MCMC posterior in both latent states and spatial loadings, whereas MFVB diverges more from the posterior but often achieves the strongest out-of-sample forecasts across 1–12 month horizons. In terms of computation, MFVB and SVB are approximately 246 times and 60 times faster than MCMC, respectively. These results underscore a practical trade-off: SVB prioritizes posterior fidelity, while MFVB maximizes speed and forecasting performance.