Conditional probability, bipartite graphs, and permutohedra

We describe a Gröbner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events, in terms of a bipartite graph. The relations may be specialized to the partially-observed random variable case, the purely conditional case, and other special cases. We also describe the connection to generalized permutohedra and describe a "conditional probability simplex".