We propose a mixed integer programming (MIP) procedure to find an outer belief approximation of a lower conditional joint cumulative distribution function (lower conditional joint CDF) obtained by the statistical matching of several sources of information, given a common variable. We assume that the variables have finite supports and we provide a procedure based on the MIP technique that produces a sparse solution with at most a given finite number of focal elements, permitting to obtain an outer approximation with a conditional belief function. In turn, the family of sparse solutions given the common variable, allows us to efficiently perform coherent inferences on new items, relying on the generalized Bayesian conditioning rule. We finally show the effectiveness of the proposed approach in the domain of company fraud detection.