Probabilistic Belief Update When Inputs Are Uncertain: The Inverse Problem Perspective

International audience ; In this paper, we examine a probabilistic intelligent system that provides posterior information, with the goal of recovering either the prior beliefs or the new data received by the system. We address this inverse problem within a general probabilistic framework and present several key contributions. Specifically, we demonstrate that when only the posterior beliefs and the data used are known, the corresponding prior beliefs form a convex set of probability distributions. We characterize this set by determining its lower and upper probability bounds. Additionally, we show that, in the general case, the minimum number of queries required to recover the prior information grows exponentially with the number of variables. We analyze this problem both when prior beliefs are represented in a non-factorized form and when they are factorized. Finally, we generalize our findings to scenarios where the available information is expressed using an alternative, non-additive theory of uncertainty.