---
title: "Cyclic Causal Models with Discrete Variables: Markov Chain Equilibrium Semantics and Sample Ordering"
authors:
  - "David Poole"
  - "Mark Crowley"
doi: ""
url: "https://ijcai.org/proceedings/2013/161"
pdf: "https://ijcai.org/proceedings/2013/0161.pdf"
venue: "Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence"
year: 2013
pages: "1060-1067"
track: ""
keywords:
---

# Cyclic Causal Models with Discrete Variables: Markov Chain Equilibrium Semantics and Sample Ordering

David Poole, Mark Crowley

*Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence*, pages 1060-1067.

## Abstract

We analyze the foundations of cyclic causal models for discrete variables, and compare structural equation models (SEMs) to an alternative semantics as the equilibrium (stationary) distribution of a Markov chain. We show under general conditions, discrete cyclic SEMs cannot have independent noise; even in the simplest case, cyclic structural equation models imply constraints on the noise. We give a formalization of an alternative Markov chain equilibrium semantics which requires not only the causal graph, but also a sample order. We show how the resulting equilibrium is a function of the sample ordering, both theoretically and empirically.