Notation for conditional distributions borrowed from statistics

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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.

Contents

Introduction

When introducing probabilistic models, it is common to see the following notation used:


\begin{align}
\xi &\sim F \\
\eta \mid \xi &\sim G(\xi).
\end{align}

By the conventions, we know that \xi and \eta are random elements in some spaces, say S and T, that F is a distribution on S and that G is a probability kernel from S to T. Each statement is a declaration about some distributional property. The first line, e.g., says that the distribution of \xi is F, and the second line states that the conditional distribution of \eta given \xi is G(\xi), 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.

Measurability

TBD.

Conditional independence and graphical models

TBD.

The notation is not a programming language

TBD.

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