Notation for conditional distributions borrowed from statistics
The following article discusses the standard notation used to declare distributional properties of collections of random variables, including subtle conventions, advanced uses, relationships with graphical models and probabilistic programs, and also common misuses and misconceptions.
When introducing probabilistic models, it is common to see the following notation used:
By the conventions, we know that and are random elements in some spaces, say and , that is a distribution on and that is a probability kernel from to . Each statement is a declaration about some distributional property. The first line, e.g., says that the distribution of is , and the second line states that the conditional distribution of given is , i.e., a random measure. In the rest of the article will discuss some subtle aspects of this notation and also point out ways that it is misused and misunderstood.
Conditional independence and graphical models
The notation is not a programming language