Relations among conditional probabilities

We describe a Gröbner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely conditional case, and other special cases. We also investigate the connection to generalized permutohedra and describe a “conditional probability simplex.” 1 Relations among conditional probabilities In 1974, Julian Besag [4] discussed the “unobvious and highly restrictive consistency conditions” among conditional probabilities. In this paper we give an answer in the discrete case to the question What conditions must a set of conditional probabilities satisfy in order to be compatible with some joint distribution? Let Ω = {1, . . . , m} be a finite set of singleton events, and let p = (p1, . . . , pm) be a probability distribution on them. Let E be a set of observable events which will be conditioned on, each a set of at least 2 singleton events. Then for events I ⊂ J , J in E , we can assign conditional probabilities for the chance of I given J , denoted pI|J . Settling Besag’s question then becomes a matter of determining the relations that must hold among the quantities pI|J . For example, Besag gives the relation (see also [3]), P (x) P (y) = n