Abstract
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.
Highlights
In recent years, interest in probabilistic programming has rapidly grown [9,11]
We study how to extend the notion of expectation transformers to conditioned probabilistic programs without non–determinism in conditional pGCL (cpGCL)
In the rest of this section we investigate some properties of the expectation transformer semantics of cpGCL
Summary
Interest in probabilistic programming has rapidly grown [9,11]. This is due to its wide applicability, for example in machine learning for describing distribution functions; Bayesian inference is pivotal in their analysis. The semantics of languages without conditioning is well–understood: In his seminal work, Kozen [19] considered denotational semantics for probabilistic programs without non–determinism or observations. One of these semantics—the expectation transformer semantics—was adopted by McIver and Morgan [21], who added support for non–determinism; a corresponding operational semantics is given in [13]. Previous work on semantics for programs with observe statements [22,16] do neither consider the possibility of non–termination nor the powerful feature of non–determinism. The observe statement blocks all invalid runs violating its condition and renormalizes the probabilities of the remaining valid runs This differs, e.g., from program annotations like (probabilistic) assertions [25] as we will see later. An extended version of this paper including all proofs and further program transformations for eliminating observe statements is available in [12]
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