Properties of Cyclic Graphical Models or Why the way up and the way down are one

 The work contained in this MS thesis seeks to extend the theory of causal models that is  elaborated in Spirtes, Glymour and Schemes’, Causation, Prediction and Search. Spirtes et. al., develop a theory of causal models, under the assumption that causation was antisymmetric, i.e. that if A is a cause of B, then B is not a cause of A. Though many causal  systems are of this kind, as the authors acknowledge in their final chapter, in many situations  one must allow for the possibility of feedback - i.e A is a cause of B, while B itself is also a  cause of A. In Chapter 1 we consider causal systems with feedback. Traditional models of systems with  feedback often assume that the system is in equilibrium, but no account is given of how the  system arrived at this state. We give a sketch of how one might attempt to give a causal  account of the process of equilibration, by interpreting an equilibrium model, as the limit  (with regard to time) of a dynamic process. In Chapter 2 we give the main result of this thesis: a computationally feasible characterization  of the conditional independencies entailed by a linear causal model with feedback. Such a  characterization will greatly facilitate the construction of efficient ‘discovery’ algorithms, that  will, under various assumptions, output an equivalence class of cyclic causal models  compatible with conditional independencies present in data taken as input. Since the  characterization itself is quite involved we try to give some insight into the intuitions that are  driving them. In addition we show how the characterization leads to a (correct) polynomial  equivalence algorithm that will decide whether two cyclic causal models entail the same sets  of conditional independencies. In Chapter 3 we prove the Cyclic Equivalence Theorem, which characterizes the cyclic  equivalence classes. Both Chapter 2 and 3 are intended to be fairly self-contained; Chapter 2  provides the level of detail necessary to understand the statement of the result, while Chapter  3 of necessity is more involved. Since there are a large number of technical defintions we  provide a Glossary is provided at the end.