depend on θ$_{i}$ , or the set of direct descendants of θ$_{i}$ in the model graph introduced in section 2.1. Finally, we denote the computational cost of evaluating the density functions corresponding to any subset x ⊂ Θ ∪ Y as dens( x ). On each iteration of Ψ$_{scalar}$ = { ψ$_{1}$, . . . , ψ$_{d}$ } , each sampler ψ$_{i}$ will incur computational cost C ( ψ$_{i}$ ) = dens( θ$_{i}$ ) + dens( x$_{i}$ ) + O (1). The trailing constant term represents generation of the proposal value and making the MH decision (generation from normal and uniform distributions, respectively). The total computational requirement of Ψ$_{scalar}$ is thus Note that under Ψ$_{scalar}$, each density evaluation dens( θ$_{i}$ ) must occur independently. This is true even when the evaluation of a particular dens( θ$_{i}$ ) term necessitates the calculation of a subsuming multivariate density – in the most extreme case, dens(Θ). Thus, in the worst case, a single MCMC iteration of Ψ$_{scalar}$ could incur d evaluations of the entire joint density of Θ. A similar computational explosion can result from the calculation of each dens( x$_{i}$ ) term. We now consider the components of C (Ψ$_{block}$). On each iteration of Ψ$_{block}$, the sole sampler u$_{block}$ (Θ) requires evaluation of the complete prior and likelihood densities, dens(Θ ∪ Y ). This is notably dif- ferent from the density evaluation terms appearing in C (Ψ$_{scalar}$), in that it incurs only a single evaluation of the complete joint model density. In addition, the adaptation routine of u$_{block}$ (Θ) requires a Cholesky factorization of the adapted covariance matrix, which requires O ( d $^{3}$) operations to calculate in full generality (Trefethen and Bau, 1997, p.176). This factorization occurs every AI iterations, where AI is the adaptation interval of u$_{block}$ , and therefore has an amortized computa- tional cost of O ( d $^{3}$/ AI). Each iteration of u$_{block}$ requires generating a d -dimensional multivariate normal proposal, which requires O ( d $^{2}$) operations, and performing a single constant-time MH de- cision, which is dropped as a lower-order term. The total amortized computational requirement of Scalar Sampling Computation Block Sampling Computation 11 C (Ψ$_{scalar}$) = d ∑ i =1 C ( ψ$_{i}$ ) = d ∑ i =1 dens( θ$_{i}$ ) + d ∑ i =1 dens( x$_{i}$ ) + O ( d ) .