  The Sobol' indices are a recognized tool in global sensitivity analysis. When
the uncertain variables in a model are statistically independent, the Sobol'
indices may be easily interpreted and utilized. However, their interpretation
and utility is more challenging with statistically dependent variables. This
article develops an approximation theoretic perspective to interpret Sobol'
indices in the presence of variable dependencies. The value of this perspective
is demonstrated in the context of dimension reduction, a common application of
the Sobol' indices. Theoretical analysis and illustrative examples are
provided.
