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 = e 2 in e 0 ) = ctx (let = e 2 in let x = e 1 in e 0 ) (provided x does not occur free in e 2 and does not occur free in e 1 ) despite the fact that e 1 and e 2 may have sampling and scoring effects.
The purpose of this case is, first, to provide students with an experience in systems modeling, using facts gathered through interviews with employees who may not be skilled in presenting their responsibilities in a systematic, logical, sequential manner. Second, students will gain actual hands-on experience learning and using a leading modeling language, the Unified Modeling Language (UML), through a popular Computer-Aided Software Engineering (C.A.S.E.) tool. Finally, the students will be using those interview facts to model an object-oriented system for processing cash receipts. In that effort, they will learn and apply the unique documentation techniques used in analyzing and designing object-oriented systems with design features such as use cases, class diagrams with inheritance, and sequence diagrams.
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