## Learning Linear Cyclic Causal Models

Inferring causal relationships from data is of fundamental importance in many areas of science. One cannot claim to have fully grasped a complex system unless one has a detailed understanding of how the different components of the system affect each other, and one cannot predict how the system will respond to some targeted intervention without such an understanding. It is well known that a statistical dependence between two measured quantities leaves the causal relation underdetermined—in addition to a causal effect from one variable to another (in either or both directions), the dependence might be due to a common cause (a confounder) of the two. In light of this underdetermination, randomized experiments have become the gold standard of causal discovery. In a randomized experiment, the values of some variable xi are assigned at random by the experimenter and, consequently, in such an experiment any correlation between xi and another measured variable xj can uniquely be attributed to a causal effect of xi on xj , since any incoming causal effect on xi (from xj , a common cause, or otherwise) would be ‘broken’ by the randomization. Since their introduction by Fisher (1935), randomized experiments now constitute an important cornerstone of experimental design. Since the 1980s causal graphical models based on directed graphs have been developed to systematically represent causal systems The causal relationships in such a model are defined in terms of stochastic functional relationships (or alternatively conditional probability distributions) that specify how the value of each variable is influenced by the values of its direct causes in the graph. In such a model, randomizing a variable xi is tantamount to removing all arrows pointing into that variable, and replacing the functional relationship (or conditional probability distribution) with the distribution specified in the experiment. The resulting truncated model captures the fact that the value of the variable in question is no longer influenced by its normal causes but instead is determined explicitly by the experimenter.

Together, the graph structure and the parameters defining the stochastic functional relationships thus determine the joint probability distribution over the full variable set under any experimental conditions. In many cases, researchers may not be willing to make some of the assumptions mentioned above, or they may want to guarantee that the full structure of the model is inferred (as opposed to only inferring an equivalence class of possible models, a common result of many discovery methods). A natural step is thus to use the power of randomized experiments. The question then becomes: Under what assumptions on the model and for what sets of experiments can one guarantee consistent learning of the underlying causal structure. The acyclicity assumption, common to most discovery algorithms, permits a straightforward interpretation of the causal model and is appropriate in some circumstances. But in many cases the assumption is clearly ill-suited. A similar situation occurs in many biological systems, where the interactions occur on a much faster time-scale than the measurements. In these cases a cyclic model provides the natural representation, and one needs to make use of causal discovery procedures that do not rely on acyclicity.

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