(right), where it displays similar attenuation as in the prior example. Most notably, we now observe the incremental effect of d diminishing as d increases, consistent with the covariance structure. Through a combination of theory and simulated examples, we observe that the algorithmic efficiency achieved under Ψ$_{scalar}$ or Ψ$_{block}$ will depend non-trivially upon the model dimension, and the extent and structure of the posterior correlation. We have only considered two simple, highly modular and systematic posterior correlation structures. In practice, any model of interest will demonstrate a substantially more complex correlation structure (which is unknown, anyway), making a full analytical study of MCMC algorithmic efficiency difficult. No less, we have only considered the two boundary-case algorithms Ψ$_{scalar}$ and Ψ$_{block}$ in Ψ$_{M}$ . Our desire to derive general results for algorithmic efficiency will be complicated substantially further when we consider the complete set Ψ$_{M}$ . While section 2.2 considered the efficiency of MCMC algorithm Ψ at producing independent sam- ples without regard for computation time, we now consider the computational requirements of Ψ, measured in units of algorithm runtime per MCMC iteration. We focus on computations for model density evaluations. There are also book-keeping and loop-iteration costs, which we label as algo- rithm overhead. These overhead terms comprise a small fraction of total computation, and we safely disregard them. Denote the computational requirement of Ψ as C (Ψ), and again overload notation to define C ( ψ ) as the computational requirement of a single sampler ψ . For Ψ = { ψ$_{1}$, . . . , ψ$_{k}$ } , C (Ψ) = ∑ k i =1 C ( ψ$_{i}$ ). As far as we are aware, an analysis of MCMC efficiency which incorporates C (Ψ) has not been carried out to date. Literature which does address MCMC efficiency typically recognizes that a computational aspect exists, but then focuses solely on A (Ψ), e.g. , Roberts and Sahu (1997). We now present an accounting of the main contributions to C (Ψ$_{scalar}$) and C (Ψ$_{block}$), general for any M . To support our accounting, we denote the set of all fixed and known components of M ( e.g., observations, other data) as Y , which is disjoint from the unknown set of parameters Θ = { θ$_{1}$, . . . , θ$_{d}$ } . For each θ$_{i}$ ∈ Θ, let x$_{i}$ ⊂ Θ ∪ Y be the set of model components which immediately 10 2.3 Computational Efficiency