blocks – the exact effect of cutting T at any particular height. We define the MCMC algorithm Ψ( T$_{cut=}$$_{h}$ ) ∈ Ψ$_{M}$ as the unique MCMC algorithm defined by scalar and/or block sampling applied to the parameter blocks that result from cutting T at height h . We note that for all T , Ψ( T$_{cut=0}$ ) = Ψ$_{scalar}$, and Ψ( T$_{cut=1}$ ) = Ψ$_{block}$. There is no universally optimal value of the cut height h , as our findings in section 2 imply that any h ∈ [0 , 1] may maximize the efficiency E (Ψ( T$_{cut=}$$_{h}$ )) for a particular model M . We recognize that a combination of distinct cut heights applied to different branches of T may produce the maximal efficiency, but we do not consider such strategies herein. Rather than attempting to infer what might be an appropriate cut height for model M , we consider a range of potential cut heights, and the resulting MCMC algorithms. These comprise Ψ$_{cand}$ , the candidate set of MCMC algorithms for a particular iteration. This approach allows the blocking procedure to adjust according to the model, the posterior correlation structure, and the underlying computational architecture. The MCMC algorithm for each iteration ( i ≥ 1) is selected from among Ψ$_{cand}$ as that which produces the maximal overall efficiency. To estimate the integrated autocorrelation time, and hence the algorithmic efficiency, of a chain of MCMC samples, we use the effectiveSize function in the R coda package (Plummer et al., 2006). The approach underlying this function – using a fitted autoregressive model to estimate the spectral density at frequency zero – has been seen to converge fastest among several methods to the true integrated autocorrelation time (Thompson, 2010). As E (Ψ$_{i}$) increases through the course of iterations, improved estimates of the posterior correla- tion are produced, giving the potential for more refined parameter blockings, and thus progressive increases in E (Ψ$_{i}$) in subsequent iterations. This iterative procedure continues until either (1) Ψ$_{i}$ = Ψ$_{i}$$_{−}$$_{1}$ (identical algorithms are selected on consecutive iterations), or (2) E (Ψ$_{i}$) < E (Ψ$_{i}$$_{−}$$_{1}$) (efficiency decreases between iterations). In practice, our procedure typically halts with terminat- ing condition (1). This may be concurrent with terminating condition (2), on account of stochastic variation in sampling and/or runtime. We select the output from our automated blocking procedure as Ψ$_{AutoBlock}$, the MCMC algorithm selected in the final iteration. In our experience, Ψ$_{AutoBlock}$ is typically identical to the MCMC algo- rithm of the second-to-last iteration; that is, the procedure has converged to a stationary state. If a 16