Causal reversibility in Bayesian networks

Causal manipulation theorems proposed by Spirtes et al. and Pearl in the context of directed probabilistic graphs, such as Bayesian networks, oŒer a simple and theoretically sound formalism for predicting the eŒect of manipulation of a system from its causal model. While the theorems are applicable to a wide variety of equilibrium causal models, they do not address the issue of reversible causal mechanisms, i.e. mechanisms that are capable of working in several directions, depending on which of their variables are manipulated exogenously. An example involving reversible causal mechanisms is the power train of a car: normally the engine moves the transmission which, in turn, moves the wheels; when the car goes down the hill, however, the driver may want to use the power train to slow down the car, i.e. let the wheels move the transmission, which then moves the engine. Some probabilistic systems can also be symmetric and reversible. For example, the noise introduced by a noisy communication channel does not usually depend on the direction of data transmission. In this paper, we investigate whether Bayesian networks are capable of representing reversible causal mechanisms. Building on the result of Druzdzel and Simon (1993), which shows that conditional probability tables in Bayesian networks can be viewed as descriptions of causal mechanisms, we study the conditions under which a conditional probability table can represent a reversible causal mechanism. Our analysis shows that conditional probability tables are capable of modelling reversible causal mechanisms but only when they ful® ll the condition of soundness, which is equivalent to injectivity in equations. While this is a rather strong condition, there exist systems where our ® nding and the resulting framework are directly applicable.