Independence and conditional independence are fundamental concepts for reasoning about groups of random variables in probabilistic programs. Verification methods for independence are still nascent, and existing methods cannot handle conditional independence. We extend the logic of bunched implications (BI) with a non-commutative conjunction and provide a model based on Markov kernels; conditional independence can be naturally expressed as a logical formula in this model. Noting that Markov kernels are Kleisli arrows for the distribution monad, we then introduce a second model based on the powerset monad and show how it can capture join dependency, a non-probabilistic analogue of conditional independence from database theory. Finally, we develop a program logic for verifying conditional independence in probabilistic programs.

中文翻译：

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依赖与独立的推理逻辑

独立性和条件独立性是推理概率程序中随机变量组的基本概念。独立性验证方法仍处于起步阶段，现有方法无法处理条件独立性。我们用非交换连接扩展了成束蕴涵 (BI) 的逻辑，并提供了一个基于马尔可夫核的模型；在这个模型中，条件独立性可以自然地表达为一个逻辑公式。注意到马尔可夫核是分布 monad 的 Kleisli 箭头，然后我们介绍了基于 powerset monad 的第二个模型，并展示了它如何捕获连接依赖，这是一种与数据库理论的条件独立性的非概率模拟。最后，我们开发了一个程序逻辑，用于验证概率程序中的条件独立性。