On the completeness of an identifiability algorithm for semi-Markovian models

Abstract

This paper addresses the problem of identifying causal effects from nonexperimental data in a causal Bayesian network, i.e., a directed acyclic graph that represents causal relationships. The identifiability question asks whether it is possible to compute the probability of some set of (effect) variables given intervention on another set of (intervention) variables, in the presence of nonobservable (i.e., hidden or latent) variables. It is well known that the answer to the question depends on the structure of the causal Bayesian network, the set of observable variables, the set of effect variables, and the set of intervention variables. Sound algorithms for identifiability have been proposed, but no complete algorithm is known. We show that the identify algorithm that Tian and Pearl defined for semi-Markovian models [1,2,3], an important special case of causal Bayesian networks, is both sound and complete. We believe that this result will prove useful to solve the identifiability question for general causal Bayesian networks.

Generic Identifiability of Linear Structural Equation Models by Ancestor Decomposition

“…Hence, the entire matrix Λ is generically identifiable, and because ( I −Λ) T Σ( I −Λ)=Ω, this implies generic identifiability of Ω. We conclude that G is generically identifiable despite being HTC‐inconclusive.Remark We remark that the idea of passing to a subgraph that is then decomposed into mixed components also appears in identification algorithms for non‐parametric structural equation models (Shpitser & Pearl, ; Huang & Valtorta, ). However, these algorithms seek to decide on global identifiability properties, whereas here we are concerned with generic identifiability.…”

The explanation game: a formal framework for interpretable machine learning

“…Specifically, some sort of adjustment must be made by conditioning on an appropriate set of covariates. While several overlapping formulations have been proposed for such adjustments (Galles and Pearl 1995;Pearl 1995;Robins 1997), we follow Tian and Pearl (2002), who provide a provably sound and complete set of causal identifiability conditions for semi-Markovian models (Huang and Valtorta 2008;Shpitser and Pearl 2008).…”

Causal Inference by String Diagram Surgery

“…Hence, it is typical to depict 2-combs in the shape of a (hair) comb, with 2 'teeth', as in (10) While combs themselves live in Stoch, Mat(R + ) accommodates a second-order reading of the transition in (10): we can treat f as a map which expects as input a map g : B 1 → A 2 and produces as output a map of type A 1 → B 2 . Plugging g : B 1 → A 2 into the 2-comb can be formally defined in Mat(R + ) by composing f and g in the usual way, then feeding the output of g into the second input of f , using caps and cups, as in (11).…”

“…Importantly, for generic f and g of Stoch, there is no guarantee that forming the composite (11) in Mat(R + ) yields a valid Stoch-morphism, i.e. a morphism satisfying the finality equation (3).…”