## Causal discovery for linear cyclic models with latent variables

Citations: | 3 - 2 self |

### BibTeX

@MISC{Hyttinen_causaldiscovery,

author = {Antti Hyttinen and Frederick Eberhardt and Patrik O. Hoyer},

title = {Causal discovery for linear cyclic models with latent variables},

year = {}

}

### OpenURL

### Abstract

We consider the problem of identifying the causal relationships among a set of variables in the presence of both feedback loops and unmeasured confounders. This is a challenging task which, for full identification, typically requires the use of randomized experiments. For linear systems, Eberhardt et al (2010) recently provided a procedure for integrating data from several experiments, and gave a corresponding, but demanding, identifiability condition. In this paper we (i) characterize the underdetermination of the model when the identifiability condition is not fully satisfied, (ii) show that their algorithm is complete with regard to the search space and the assumptions, and (iii) extend the procedure to incorporate the common assumption of faithfulness, and any prior knowledge. The resulting method typically resolves much additional structure and often yields full identification with many fewer experiments. We demonstrate our procedure using simulated data, and apply it to the protein signaling dataset of Sachs et al (2005). 1

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Citation Context ...ata, while allowing both feedback loops and confounding hidden variables. We consider a standard class of models known as linear non-recursive structural equation models with correlated disturbances (=-=Bollen, 1989-=-). Specifically, let V = {x1, . . . , xN} denote the set of observed variables. Arranging these variables into the vector x, the linear model is given by x := Bx + e, (1) where each element b ji of B ... |

520 |
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Citation Context ... causes of the set of variables under consideration, or when there are feedback loops. Consequently, constraints are typically placed on the search space the algorithms consider: The ‘FCI’ algorithm (=-=Spirtes et al., 2000-=-) only considers acyclic causal structures but allows latent variables, while the ‘CCD’ algorithm of Richardson (1996) can handle cyclic causal systems but does not allow for latents. Even with these ... |

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62 | Causal discovery from a mixture of experimental and observational data
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(Show Context)
Citation Context ... (Tillman et al., 2009; Triantafillou et al., 2010; Tillman and Spirtes, 2011). On the other hand there are several procedures that combine multiple experimental studies on the same set of variables (=-=Cooper and Yoo, 1999-=-; Tong and Koller, 2001; Murphy, 2001; Eaton and Murphy, 2007; He and Geng, 2008; Schmidt and Murphy, 2009; Eberhardt et al., 2010; Hyttinen et al., 2010). In this article we present methods that comb... |

56 | 2001b. Active learning for structure in Bayesian networks
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(Show Context)
Citation Context ...; Triantafillou et al., 2010; Tillman and Spirtes, 2011). On the other hand there are several procedures that combine multiple experimental studies on the same set of variables (Cooper and Yoo, 1999; =-=Tong and Koller, 2001-=-; Murphy, 2001; Eaton and Murphy, 2007; He and Geng, 2008; Schmidt and Murphy, 2009; Eberhardt et al., 2010; Hyttinen et al., 2010). In this article we present methods that combine the two approaches:... |

42 | Active learning of causal bayes net structure
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(Show Context)
Citation Context ... 2010; Tillman and Spirtes, 2011). On the other hand there are several procedures that combine multiple experimental studies on the same set of variables (Cooper and Yoo, 1999; Tong and Koller, 2001; =-=Murphy, 2001-=-; Eaton and Murphy, 2007; He and Geng, 2008; Schmidt and Murphy, 2009; Eberhardt et al., 2010; Hyttinen et al., 2010). In this article we present methods that combine the two approaches: we consider d... |

27 | Exact Bayesian structure learning from uncertain interventions
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(Show Context)
Citation Context ... and Spirtes, 2011). On the other hand there are several procedures that combine multiple experimental studies on the same set of variables (Cooper and Yoo, 1999; Tong and Koller, 2001; Murphy, 2001; =-=Eaton and Murphy, 2007-=-; He and Geng, 2008; Schmidt and Murphy, 2009; Eberhardt et al., 2010; Hyttinen et al., 2010). In this article we present methods that combine the two approaches: we consider data sets that contain pa... |

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Citation Context ...l results are needed to complement any available ones. Note that K specifically chosen experiments are enough to the satisfy the identifiability condition for models with up to ( ) K ⌊K/2⌋ variables (=-=Spencer, 1970-=-). 4 FAITHFULNESS & BILINEARITY In most realistic contexts the set of available overlapping data sets will not contain a sufficient variety of experimental interventions to satisfy the demanding ident... |

8 | Modeling discrete interventional data using directed cyclic graphical models - Schmidt, Murphy - 2009 |

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(Show Context)
Citation Context ...the other hand there are several procedures that combine multiple experimental studies on the same set of variables (Cooper and Yoo, 1999; Tong and Koller, 2001; Murphy, 2001; Eaton and Murphy, 2007; =-=He and Geng, 2008-=-; Schmidt and Murphy, 2009; Eberhardt et al., 2010; Hyttinen et al., 2010). In this article we present methods that combine the two approaches: we consider data sets that contain passive observational... |

5 | Combining experiments to discover linear cyclic models with latent variables
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(Show Context)
Citation Context ...s that all independencies in the data are derived from the structure of the graph, rather than specific parameter values (Spirtes et al., 2000; Pearl, 2000). For maximum generality, the algorithm in (=-=Eberhardt et al., 2010-=-) did not use the assumption of faithfulness. However, given the demanding identifiability conditions (see Section 1), it is worth investigating whether faithfulness might add substantial benefit when... |

5 | Systems of bilinear equations
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(Show Context)
Citation Context ...olves characterizing the solution set for such a bilinear equation system. Unfortunately, no efficient solution methods for large bilinear equation systems are known, except in certain special cases (=-=Cohen and Tomasi, 1997-=-; Johnson and Link, 2009). Nevertheless, one can attempt to solve the system byminimizing the objective function C(B, T) = ‖K1vec(B) − k1‖ 2 + (4) ‖K2vec(T) − k2‖ 2 + ‖T(I − B) − I‖ 2 , where the ‖ ·... |

4 |
Feedback Models: Interpretation and Discovery
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(Show Context)
Citation Context ...vide orientation rules for search spaces where the true causal structure either contains latent variables but no cycles (FCI, (Spirtes et al., 2000)) or contains cycles but no latent variables (CCD, (=-=Richardson, 1996-=-)). Since our model space contains both latent variables and cycles, and we have the advantage of experiments, different orientation rules are required. We employ the following two rules that take adv... |

3 | Structure learning in causal cyclic networks - Itani, Ohannessian, et al. - 2008 |

3 | A logical characterization of constraint-based causal discovery - Claassen, Heskes - 2011 |

2 |
Learning equivalence classes of acyclic models with latent and selection variables from multiple datasets with overlapping variables
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(Show Context)
Citation Context ...nt ways. On the one hand a variety of algorithms have been developed that integrate data from overlapping but purely passive observational data sets (Tillman et al., 2009; Triantafillou et al., 2010; =-=Tillman and Spirtes, 2011-=-). On the other hand there are several procedures that combine multiple experimental studies on the same set of variables (Cooper and Yoo, 1999; Tong and Koller, 2001; Murphy, 2001; Eaton and Murphy, ... |

1 | Learning linear cyclic causal models with latent variables. Submitted. Available online from the authors’ homepages
- Hyttinen, Eberhardt, et al.
- 2012
(Show Context)
Citation Context ...ments of B, related to self-loops (edges from a node to itself) in the model. These do not affect the equilibrium distribution in linear cyclic models, and can be handled by a suitable normalization (=-=Hyttinen et al., 2012-=-). 2 A detailed formal account of the intervened model is given by Hyttinen et al. (2012), see their Equation 7 and Lemma 5.case of overlapping data sets we have to make a slightly stronger assumptio... |

1 |
Solution theory for complete bilinear systems of equations. Numerical Linear Algebra with Applications
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- 2009
(Show Context)
Citation Context ... solution set for such a bilinear equation system. Unfortunately, no efficient solution methods for large bilinear equation systems are known, except in certain special cases (Cohen and Tomasi, 1997; =-=Johnson and Link, 2009-=-). Nevertheless, one can attempt to solve the system byminimizing the objective function C(B, T) = ‖K1vec(B) − k1‖ 2 + (4) ‖K2vec(T) − k2‖ 2 + ‖T(I − B) − I‖ 2 , where the ‖ · ‖2 denotes squared Eucl... |