Prioritization of Variables

Schervish (1984) originally suggested that the computation of $P({\bf b})$ should be easier for his numerical integration method if the variables are reordered (and appropriate rows and columns of $\Sigma$are permuted) so that the largest b_i values are limits for the innermost integrals. This sorting can also make the numerical integration easier after Genz's transformations because it can cause the innermost integrals to have expected value closer to one and can therefore reduce the overall variation in the integrand. Genz (1992) included this sorting in his implementations. This sorting is also expected to reduce the time taken by acceptance-rejection sampling. With this method, the test for rejection of a point ${\bf y}_j$is done by computing the components $\{x_i\}$ of $C{\bf y}_j$ in sequence, and rejecting ${\bf y}_j$ as soon as x_i > b_i, for some i. The rejection of a point ${\bf y}_j$ is likely to be detected sooner (avoiding computation of the rest of the components of $C{\bf y}_j$) when the b_i's are sorted.

Gibson, Glasbey and Elston (1992), who independently developed a Monte-Carlo method similar to the one developed by Genz, suggested an improved prioritization of the variables. With this technique, the variables are sorted so that the innermost integrals have the largest expected integration intervals. This is more complicated than just sorting the integration limits and permuting the respective rows and columns of $\Sigma$, because the Cholesky factor C must be computed dynamically during the sorting of the variables. But this method uses both ${\bf b}$ and $\Sigma$ in the sorting process, it should therefore further increases the likelihood that the innermost integrals have values close to one and improve the convergence of the numerical integration methods. The overall cost (O(m³)) is not significant compared to the rest of the computation cost for the methods discussed here, and is therefore a relatively cheap preconditioning step that can be used with the algorithms. This sorting technique was used in the implementations of the algorithms for the tests reported in the next section.