On aggregating probabilistic evidence

Abstract

Abstract Imagine a database—a set of propositions $\varGamma =\{F_1,\ldots ,F_n\}$ with some kind of probability estimates and let a proposition $X$ logically follow from $\varGamma $. What is the best justified lower bound of the probability of $X$? The traditional approach, e.g. within Adams’ probability logic, computes the numeric lower bound for $X$ corresponding to the worst-case scenario. We suggest a more flexible parameterized approach by assuming probability events $u_1,u_2,\ldots ,u_n$ that support $\varGamma $ and calculating aggregated evidence$e(u_1,u_2,\ldots ,u_n)$ for $X$. The probability of $e$ provides a tight lower bound for any, not only a worst-case, situation. The problem is formalized in a version of justification logic and the conclusions are supported by corresponding completeness theorems. This approach can handle conflicting and inconsistent data and allows the gathering both positive and negative evidence for the same proposition.