Abstraction is a powerful idea widely used in science, to model, reason and explain the behavior of systems in a more tractable search space, by omitting irrelevant details. While notions of abstraction have matured for deterministic systems, the case for abstracting probabilistic models is not yet fully understood.

In this paper, we provide a semantical framework for analyzing such abstractions from first principles. We develop the framework in a general way, allowing for expressive languages, including logic-based ones that admit relational, deterministic and hierarchical constructs with stochastic primitives. We motivate a definition of consistency between a high-level model and its low-level counterpart, but also treat the case when the high-level model is missing critical information present in the low-level model. We go on to prove properties of abstractions, both at the level of the parameter as well as the structure of the models. We conclude with some observations about how abstractions can be derived automatically.

摘要是一种在科学中广泛使用的有力思想，它通过省略无关的细节来在更易处理的搜索空间中对系统的行为进行建模，推理和解释。尽管确定性系统的抽象概念已经成熟，但是对概率模型抽象的情况尚未完全理解。

在本文中，我们提供了一个语义框架，用于从第一原理分析此类抽象。我们以一种通用的方式来开发框架，允许使用表达性语言，包括基于逻辑的语言，这些语言接受具有随机基元的关系，确定性和层次结构。我们激励定义高级模型与低级模型之间的一致性，但也要处理高级模型缺少低级模型中存在的关键信息的情况。我们继续在参数级别以及模型结构上证明抽象的属性。我们以关于如何自动派生抽象的一些结论结束。