Problems Related to Causal Reasoning in Equilibrium Models
Causal discovery, for the most part, is concerned with learning causal models in the form of directed acyclic graphs (DAGs) from equilibrium (as opposed to time series) data. Causal reasoning, by contrast, is concerned with using such causal DAGs to perform inferences. In particular, much work on causal reasoning has focused on the ability to predict the new probability distribution over a set of variables, V , given a causal graph G = (V; E) and given the fact that some subset of variables V 0 V has been externally manipulated to some conguration. These types of manipulation inferences contrast with more common diagnostic inferences, in that the former may require the causal graph to be altered prior to performing probabilistic inference. Specically , the ability to perform manipulation inferences is made possible by a critical postulate which we call the Manipulation Postulate. All formalisms for causal reasoning take the manipulation postulate as a fundamental starting point: The Manipulation Postulate. If G = (V; E) is a causal graph and V 0 V is a subset of variables being manipulated, then the causal graph, G 0 , describing the manipulated system is such that G 0 = (V; E 0 ), where E 0 E and E 0 diers from E by at most the set of arcs into V 0 . In other words, manipulating a variable can cause some of its incoming arcs to be removed from the causal graph, but can eect no other change in the causal graph. The Manipulation Theorem of Spirtes, et. al. (1992) proves that given the Manipulation Postulate and the Markov Condition, the probability distribution of the manipulated model can be calculated. Furthermore, the axiomatizations of causal reasoning of Galles and Pearl (1997) and of Halpern (1998) also take the Manipulation Postulate as a fundamental assumption. The question that we pose in this paper is \Are these two lines of research (i.e., equilibrium causal discovery and manipulation reasoning) consistent?" Namely, what would happen if we took an equilibrium causal model (learned from data), and applied the manipulation formalisms to it? Are the resulting inferences guaranteed to be valid? We prove by explicit counterexample that such inferences are not guaranteed to be valid in the sense that conditional independencies in the manipulated model can dier from the conditional independencies in the learned model of the manipulated system. Symbolically, if MS is a learned causal model of system S, and if we use the^operator to denote manipulation, then we show that ^ MS 6 M ^ S . Our general strategy is as follows: We rst present two extremely simple physical systems (an ideal gas trapped in a cylinder with a movable piston and a mass dangling from a damped spring), we show, based on physical laws what the \true" equilibrium causal graphs of these systems are.