  The constraint of a progressive decrease in residual renormalization scale
dependence with increasing loop order is developed as a method for obtaining
bounds on unknown higher-order perturbative corrections to
renormalization-group invariant quantities. This technique is applied to the
inclusive semileptonic process $b\to u \bar\nu_\ell\ell^-$ (explicitly known to
two-loop order) to obtain bounds on the three- and four-loop perturbative
coefficients that are not accessible via the renormalization group. Using the
principle of minimal sensitivity, an estimate is obtained for the perturbative
contributions to $\Gamma(b\to u \bar\nu_\ell\ell^-)$ that incorporates
theoretical uncertainty from as-yet-undetermined higher order QCD corrections.
