Many aspects of social welfare are intrinsically multidimensional. Composite indices at-tempting to reduce this complexity to a unique measure abound in many areas of economics and public policy. Comparisons based on such measures depend, sometimes critically, on how the different dimensions of performance are weighted. Thus, a policy maker may wish to take into account imprecision over composite index weights in a systematic manner. In this paper, such weight imprecision is parameterized via the e-contamination framework of Bayesian statistics. Subsequently, combining results from polyhedral geometry, social choice, and theoretical computer science, an analytical procedure is presented that yields a provably robust ranking of the relevant alternatives in the presence of weight imprecision. The main idea is to consider a vector of weights as a voter and a continuum of weights as an electorate. The procedure is illustrated on recent versions of the Rule of Law and Human Development indices.