# Contextual Equivalence for a Probabilistic Language with Continuous Random Variables and Recursion

We present a complete reasoning principle for contextual equivalence in an untyped probabilistic language. The language includes continuous (real-valued) random variables, conditionals, and scoring. It also includes recursion, since the standard call-by-value fixpoint combinator is expressible. We demonstrate the usability of our characterization by proving several equivalence schemas, including familiar facts from lambda calculus as well as results specific to probabilistic programming. In particular, we use it to prove that reordering the random draws in a probabilistic program preserves contextual equivalence. This allows us to show, for example, that (let x = e_1 in let y = e_2 in e_0) is equivalent to (let y = e_2 in let x = e_1 in e_0) (provided x does not occur free in e_2 and y does not occur free in e_1) despite the fact that e_1 and e_2 may have sampling and scoring effects.

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