Conditioning in Probabilistic Programming

This article investigates the semantic intricacies of conditioning, a main feature in probabilistic programming. Our study is based on an extension of the imperative probabilistic guarded command language pGCL with conditioning. We provide a weakest precondition (wp) semantics and an operational semantics. To deal with possibly diverging program behavior, we consider liberal preconditions. We show that diverging program behavior plays a key role when defining conditioning. We establish that weakest preconditions coincide with conditional expected rewards in Markov chains—the operational semantics—and that the wp-semantics conservatively extends the existing semantics of pGCL (without conditioning). An extension of these results with nondeterminism turns out to be problematic: although an operational semantics using Markov decision processes is rather straightforward, we show that providing an inductive wp-semantics in this setting is impossible. Finally, we present two program transformations that eliminate conditioning from any program. The first transformation hoists conditioning while updating the probabilistic choices in the program, while the second transformation replaces conditioning—in the same vein as rejection sampling—by a program with loops. In addition, we present a last program transformation that replaces an independent identically distributed loop with conditioning.