nference methods are often formulated as variational approximations: these approxima-tions allow easy evaluation of statistics by marginalization or linear response, but theseestimates can be inconsistent. We show that by introducing constraints on covariance, onecan ensure consistency of linear response with the variational parameters, and in so doinginference of marginal probability distributions is improved. For the Bethe approximationand its generalizations, improvements are achieved with simple choices of the constraints.The approximations are presented as variational frameworks; iterative procedures relatedto message passing are provided for finding the minim
