I will present a semantic foundation for probabilistic programming languages that support higher-order functions, continuous distributions, and soft constraints. This provides a solid foundation for justifying program transformations and inference algorithms. From another perspective, it allows us to understand basic facts from probability in terms of program equations. 
The work is based on a paper presented at LICS 2016 (arXiv:1601.04943 , jointly with Yang, Staton, Kammar and Wood), with some important new developments.

Video Recording