We now study the potential for algorithmic inefficiency that may result from scalar or block sampling, by examining situations in which each are particularly ill-suited. Consider MCMC algorithm Ψ$_{block}$ ∈ Ψ$_{M}$ . Application of Ψ$_{block}$ generates a random proposal vector X ∗ ∈ R $^{d}$, then jointly accepts or rejects X $^{∗}$. In the framework of the sampling method u$_{block}$ , X ∗ ∼ N$_{d}$( µ, σ 2 $_{d}$Σ), where µ and Σ vary according to current state of the MCMC chain and properties of the target stationary distribution, but not proportionally to d . Roberts, Gelman, and Gilks (1997) show that in order to achieve the asymptotically optimal acceptance rate, and therefore sample chain mixing, σ 2 d ∝ d − $^{1}$. As a consequence of this attenuation in the proposal variance, the rate at which Ψ$_{block}$ explores the space of R $^{d}$, and accordingly A (Ψ$_{block}$), is proportional to d − $^{1}$. This result applies equivalently to block samplers ψ = u$_{block}$ ( b ) acting on subsets b ⊂ Θ, where the algorithmic efficiency (for the elements of b ) achieved by application of ψ is inversely proportional to the number of elements in b . This result has an important implication on block sampling. All other factors being equal ( e.g. , effect of posterior correlations), a block sampler of dimension k must generate k times more samples to have the same effective sample size as those samples produced through scalar sampling (Roberts and Rosenthal, 2001). This inefficiency is inherent to block sampling and scales with the dimension of any block sampler. To study the potential loss of algorithmic efficiency which may result from scalar sampling under Ψ$_{scalar}$ ∈ Ψ$_{M}$ , we consider correlated posterior distributions. It is well-understood that strong posterior correlations can retard the speed of convergence of MCMC sampling ( e.g., Roberts and Sahu, 1997; Gilks, 2005), and that block sampling can alleviate this. However, the nature of this inefficiency is not characterized, in particular as a function of the degree of correlation and number of dimensions exhibiting correlation. We undertake a simulation study, to gauge how these factors effect algorithmic efficiency. Consider target distribution N$_{d}$(0 , Σ), where the covariance (equivalently, correlation) matrix Σ consists of 1s on the main diagonal and ρ elsewhere, | ρ | < 1. 8 Efficiency loss from scalar sampling Efficiency loss from block sampling