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Standard approaches to probabilistic reasoning require that one possesses an explicit model of the distribution in question. But, the empirical learning of models of probability distributions from partial observations is a problem for which efficient algorithms are generally not known. In this work we consider the use of bounded-degree fragments of the "sum-of-squares" logic as a probability logic. Prior work has shown that we can decide refutability for such fragments in polynomial-time. We propose to use such fragments to answer queries about whether a given probability distribution satisfies a given system of constraints and bounds on expected values. We show that in answering such queries, such constraints and bounds can be implicitly learned from partial observations in polynomial-time as well. It is known that this logic is capable of deriving many bounds that are useful in probabilistic analysis. We show here that it furthermore captures useful polynomial-time fragments of resolution. Thus, these fragments are also quite expressive.

Polynomial-Time Probabilistic Reasoning with Partial Observations via Implicit Learning in Probability Logics