  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.
