Ψ$_{block}$ may be written as We have seen that C (Ψ$_{block}$) is at least quadratic in d , and technically cubic but with a small lead- ing coefficient. Depending on the distributional structure of Θ, the density evaluations comprising C (Ψ$_{scalar}$) may be unwieldy. The relative magnitude of these competing terms is difficult to intu- itively gauge, so to gain practical insight, we perform a timing study of the Ψ$_{scalar}$ (All Scalar) and Ψ$_{block}$ (All Blocked) algorithms. Three models involving no likelihood components are considered, with prior structures on Θ given as: • θ$_{i}$ ∼ N( µ, σ ) for each θ$_{i}$ ∈ Θ • θ$_{i}$ ∼ Gamma( α, β ) for each θ$_{i}$ ∈ Θ • Θ ∼ N$_{d}$( µ, Σ) Figure 2 presents timing results measured in seconds per 10,000 iterations, plotted against di- mension d , without consideration of algorithmic efficiency (section 2.2). There are a number of inter- esting features, which we briefly summarize. C (Ψ$_{scalar}$) is O ( d ) when each θ$_{i}$ independently follows a univariate distribution, and therefore ∑ d i $_{=1}$dens( θ$_{i}$ ) = dens(Θ) ≤ d · K , where K = max$_{θ}$$_{∈}$$_{Θ}$ dens( θ ). For all practical purposes, it appears that C (Ψ$_{block}$) is O ( d $^{2}$), as the cubic coefficient 1 / AI = 1 / 200 is relatively small. C (Ψ$_{block}$) is largely resilient to changes in the underlying distribution of Θ; univariate gamma densities are more costly than normal densities, as we would expect, and as for C (Ψ$_{scalar}$); and the multivariate normal structure even slightly more so. Perhaps most striking, C (Ψ$_{scalar}$) is O ( d $^{3}$) when the underlying distribution of Θ is multivariate, since each multivariate normal density evaluation is O ( d $^{2}$), which occurs d times for each iteration of Ψ$_{scalar}$. Although both are cubic in d , C (Ψ$_{scalar}$) dwarfs C (Ψ$_{block}$) due to the difference in the leading cubic coefficients. 12 C (Ψ$_{block}$) = dens(Θ ∪ Y ) + O ( d $^{3}$/ AI) + O ( d $^{2}$) . Timing Comparison