Conditioning in Probabilistic Programming

In this paper, we investigate the semantic intricacies of conditioning in probabilistic programming, a major feature, e.g., in machine learning. We provide a quantitative weakest pre–condition semantics. In contrast to all other approaches, non–termination is taken into account by our semantics. We also present an operational semantics in terms of Markov models and show that expected rewards coincide with quantitative pre–conditions. A program transformation that entirely eliminates conditioning from programs is given; the correctness is shown using our semantics. Finally, we show that an inductive semantics for conditioning in non–deterministic probabilistic programs cannot exist.