We give an adequate denotational semantics for languages with recursive higher-order types, continuous probability distributions, and soft constraints. These are expressive languages for building Bayesian models of the kinds used in computational statistics and machine learning. Among them are untyped languages, similar to Church and WebPPL, because our semantics allows recursive mixed-variance datatypes. Our semantics justifies important program equivalences including commutativity. Our new semantic model is based on `quasi-Borel predomains'. These are a mixture of chain-complete partial orders (cpos) and quasi-Borel spaces. Quasi-Borel spaces are a recent model of probability theory that focuses on sets of admissible random elements. Probability is traditionally treated in cpo models using probabilistic powerdomains, but these are not known to be commutative on any class of cpos with higher order functions. By contrast, quasi-Borel predomains do support both a commutative probabilistic powerdomain and higher-order functions. As we show, quasi-Borel predomains form both a model of Fiore's axiomatic domain theory and a model of Kock's synthetic measure theory.

翻译：我们为循环性较高排序类型、连续概率分布和软约束的语言给出适当的省略语义。 这些语言是用于构建计算统计和机器学习中使用的巴伊西亚模型的表达语言。 其中包括非类型语言, 类似于教堂和WebPPL, 因为我们的语义允许循环性混合变量数据类型。 我们的语义为包括通电在内的重要程序等同提供了理由。 我们的新语义模型基于“ quasi- borel predomains ” 。 这些是链状部分命令( cpos) 和准布尔空间的混合体。 Quasi- Borel空间是近期的概率理论模型模型, 侧重于可允许随机元素的组合。 概率传统上在 cpo 模型中处理, 使用概率性混合性功能, 但人们并不知道这些功能更高排序的模型具有共通性。 相比之下, 准Boel presmais 和高级合成理论性模型的模型函数。 我们展示了一种准稳定性模型和高级理论的模型。