A universal probabilistic programming language consists of a general-purpose language extended with two probabilistic features: the ability to make random (probabilistic) choices and the ability to make observations. For expressiveness and efficiency, it is useful to consider observations that have nonnegative real likelihoods rather than simple boolean truth values. A program in such a language represents a probabilistic process; the chance of producing a particular answer is determined by the random choices made along the way and the likelihoods of the observations.

Existing probabilistic programming languages typically support a constrained form of observation—such as requiring the distribution in explicit form—or a general weighting operation, like factor. This work explores the interaction between observation and the other computational features of the language. We present a big-step semantics of importance sampling with likelihood weighting for a core universal probabilistic programming language with observation propagation.