We consider a Bayesian implementation of a new approach to estimating Demand Systems. This approach, suggested by Varian (1990), is based on a generalization of Afriat's (1967) e-ciency index. The model we propose leads to a very tractable posterior and predictive analysis, yet allows for interesting economic interpretations. We conduct a sensitivity analysis with respect to the prior in an application to annual aggregate U.S. consumption data, and conclude that the sample is quite informative. Average e-ciency and expected budget shares are examined in some detail.