Directed Cyclic Graphical Representations of Feedback Models

By Peter Spirtes. 1995

The use of directed acyclic graphs (DAGs) to represent conditional independence relations among random variables has proved fruitful in a variety of ways. Recursive structural equation models are one kind of DAG model. However, non-recursive structural equation models of the kinds used to model economic processes are naturally represented by directeed cyclic graphs (DCG). For linear systems associated with DCGs with independent errors, a characterisation of conditional independence constraints is obtained, and it is shown that the result generalizes in a natural way to systems in which the error variables or noises are statistically dependent. For non-linear systems with independent errors a sufficient condition for conditional independence of variables in associated distributions is obtained.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

From Deterministic ODEs to Dynamic Structural Causal Models (2016, background): “One of the successes of SCMs over other causal frameworks such as causal Bayesian networks, for instance, has been their ability to express cyclic causal models (Spirtes, 1995; Mooij et al., 2011; Hyttinen et al., 2012; Voortman et al., 2010; Lacerda et al., 2008; Bongers et al., 2018).”
From Ordinary Differential Equations to Structural Causal Models: the deterministic case (2013, methodology): “Over the years, several attempts have been made to extend the theory to the cyclic case, thereby enabling causal modeling of systems that involve feedback (Spirtes, 1995; Koster, 1996; Pearl and Dechter, 1996; Neal, 2000; Hyttinen et al., 2012).”
CAM: Causal Additive Models, high-dimensional order search and penalized regression (2013, background): “Finally, it would be very interesting to extend (sparse) permutation search to (possibly nonidentifiable) models with hidden variables [7, 11, 23, 34] or with graph structures allowing for cycles [19, 20, 30, 33].”
Foundations of structural causal models with cycles and latent variables (2016, background): “It does not hold in general for cyclic SCMs, however, as was already observed earlier [66, 67].”
On Causal Discovery with Cyclic Additive Noise Models (2011, background): “Although the linear case has been studied in considerable detail already [6, 7, 8], as far as we know, nobody has yet investigated the (more realistic) case of nonlinear causal mechanisms.”
A Discovery Algorithm for Directed Cyclic Graphs (1996, background): “The independencies encoded by a given graph are determined by a graphical criterion called d-separation, as explained for the acyclic case in Pearl (1988), and extended to the cyclic case in Spirtes (1995) (See also Koster(1994)).”
Discovering Cyclic Causal Models with Latent Variables: A General SAT-Based Procedure (2013, background): “The correspondence is known to hold for linear causal relations with Gaussian error terms, i.e. non-recursive linear Gaussian structural equation models (Spirtes, 1995), and can be extended to models with correlated error terms, which is one way to account for causally insuﬃcient sets of variables.”
Partial correlation for functional brain interactivity investigation in functional MRI (2006, background): “These problems, which are critical for directed acyclic graphs, are even more crucial for cyclic graphs, which are commonly used in effective connectivity modeling, while many theoretical questions remain unsolved (Spirtes, 1995; Pearl and Dechter, 1996; Neal, 2000).”
Ancestral graph Markov models (2002, methodology): “…the following:Proposition 8.24 If G is an an estral graph ontaining no undire ted edgesthen N (G) is the model obtained by regarding G as a path diagram.69 Further results relating path diagrams and graphi al models are des ribedin Spirtes et al. (1998), Koster (1999a,b, 1996) and Spirtes (1995).”
Joint Causal Inference from Multiple Contexts (2016, methodology): “The first one to address this using constraint-based causal discovery was Tillman (2009), and several other methods have been proposed over the years (Claassen and Heskes, 2010; Tillman and Spirtes, 2011; Hyttinen et al.”
Caveats for Causal Reasoning with Equilibrium Models (2001, background): “because although cyclic graphs may obey the Markov condition locally, they will not in general obey this condition in a global sense [Spirtes, 1995], which makes the factorization in Equation A.”
Review of Causal Discovery Methods Based on Graphical Models (2019, background): “For linear systems, the graphical property of d -separation has been generalized to directed graphs with cycles–closed directed paths (Spirtes, 1995).”
Causal Pluralism versus Epistemic Causality (2006, methodology): “…that causality is slightly more complicated than that – e.g., Schaffer (2005) argues that it is not binary, Mellor (1995) that it is not a relation, Spirtes (1995) that it is not acyclic – but these alternative conceptions are still simple enough to make the subsequent point, and the analysis of…”
Graphical models for marked point processes based on local independence (2007, background): “systems (Spirtes, 1995; Koster, 1996) but are still based on cross-sectional data.”
Learning linear cyclic causal models with latent variables (2012, unknown): “…break any incoming arrows on the intervened variables, this independence also follows directly from the graphical d-separation criterion extended to cyclic graphs (Spirtes, 1995): In Figure 3, variables x1 and x3 are not d-connected by any of the undirected paths through the double headed arrows.”