Exchangeable Random Processes and Data Abstraction

Several Turing-complete probabilistic programming languages such as Church, Venture and Anglican support so-called exchangeable random processes (XRPs) as built-in abstract data types. Instances of these data types denote infinite sequences of random values. Every such object satisfies the so-called exchangeability property, which says that the probability distribution of the object does not change under permutations of the sequence. In fact, exchangeability is one of the core concepts in Bayesian statistics, from which these infinite sequences originate. (Extensions of the notion of exchangeability for sequences apply to other infinite random objects such as graphs, which we consider in Section 3.) We have been working on semantic foundations and reasoning principles for data types for exchangeable random processes, with special emphasis on the client programs of these data types. One slogan that summarises our efforts well is this: a data type for a random process is exchangeable if its inclusion in a probabilistic programming language does not break the following commutativity and discardability of the language, i.e.,