# On Finite Exchangeability and Conditional Independence

## TL;DR

This paper characterizes the independence structures of finitely exchangeable distributions over vectors and networks, identifying conditions for independence and dependence, and exploring complex regimes in exchangeable networks.

## Contribution

It provides necessary and sufficient conditions for independence in exchangeable vectors and extends these results to complex regimes in exchangeable networks.

## Key findings

- Conditions for complete independence in exchangeable vectors
- Conditions for complete dependence in exchangeable vectors
- Six dual regimes of independence in exchangeable networks

## Abstract

We study the independence structure of finitely exchangeable distributions over random vectors and random networks. In particular, we provide necessary and sufficient conditions for an exchangeable vector so that its elements are completely independent or completely dependent. We also provide a sufficient condition for an exchangeable vector so that its elements are marginally independent. We then generalize these results and conditions for exchangeable random networks. In this case, it is demonstrated that the situation is more complex. We show that the independence structure of exchangeable random networks lies in one of six regimes that are two-fold dual to one another, represented by undirected and bidirected independence graphs in graphical model sense with graphs that are complement of each other. In addition, under certain additional assumptions, we provide necessary and sufficient conditions for the exchangeable network distributions to be faithful to each of these graphs.