second-order AR process) relative to All Blocked sampling. In addition to the improved Runtime, the All Blocked sampling of the correlated AR process values increases their individual algorithmic efficiencies, and the slowest mixing parameter is among the fixed effects. The Efficiency under All Blocked sampling is over double that of All Scalar sampling. The automated blocking procedure identifies a blocking scheme which blocks together all AR process values and fixed effects (23 total; cut height h = 0.4), and performs univariate sampling of the single hyper-parameter. This has a similar Runtime to All Blocked sampling, but increases algorithmic efficiency for all parameters. The resulting overall Efficiency under the Auto Blocking MCMC algorithm is over three times that of All Scalar sampling. Table 1: MCMC performance results for the suite of example models. Effective sample size (ESS) is measured in effective samples per 10,000 iterations, Runtime is presented as seconds per 10,000 iterations, and Efficiency is in units of effective samples produced per second of algorithm runtime. Model MCMC Scheme ESS Runtime Efficiency Random Effects All Blocked 0.4 0.29 1.3 Default 1.1 1.19 1.0 All Scalar 2.1 0.51 4.2 Informed Blocking 19.0 0.50 38.2 Informed Cross-Level 101.3 2.64 38.5 Auto Blocking 19.0 0.48 39.2 Auto- Regressive All Blocked 8.9 0.3 27.3 All Scalar 6.5 0.6 11.5 Auto Blocking 12.7 0.3 37.5 State Space Independent All Blocked 0.3 0.8 0.4 Default 27.6 4.6 6.0 All Scalar 20.2 1.3 15.7 Auto Blocking 29.1 1.3 22.4 State Space Correlated All Blocked 0.6 0.7 0.8 Default 1.7 4.9 0.4 All Scalar 1.1 1.3 0.8 Informed Blocking 18.4 1.2 15.6 Auto Blocking 26.1 1.2 20.9 Spatial All Blocked 0.2 5.71 0.04 Default 0.4 10.86 0.04 All Scalar 171.3 83.87 2.0 Auto Blocking 1208.0 78.62 15.4 GLMM All Blocked 2.2 44.3 0.05 All Scalar 60.9 22.6 3.0 Auto Blocking 60.9 22.6 3.0 24