We consider a decentralized detection network whose aim is to infer a public hypothesis of interest. However, the raw sensor observations also allow the fusion center to infer private hypotheses that we wish to protect. We consider the case where there are an uncountable number of private hypotheses belonging to an uncertainty set, and develop local privacy mappings at every sensor so that the sanitized sensor information minimizes the Bayes error of detecting the public hypothesis at the fusion center, while achieving information privacy for all private hypotheses. We introduce the concept of a most favorable hypothesis (MFH) and show how to find a MFH in the set of private hypotheses. By protecting the information privacy of the MFH, information privacy for every other private hypothesis is also achieved. We provide an iterative algorithm to find the optimal local privacy mappings, and derive some theoretical properties of these privacy mappings. Simulation results demonstrate that our proposed approach allows the fusion center to infer the public hypothesis with low error while protecting information privacy of all the private hypotheses.