Determinantal Generalizations of Instrumental Variables

Abstract

Abstract. Linear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the random vector. The graph contains directed edges that represent the linear relationships between components, and bidirected edges that encode unobserved confounding. We study the problem of generic identifiability, that is, whether a generic choice of linear and confounding effects can be uniquely recovered from the joint covariance matrix of the observed random vector. An existing combinatorial criterion for establishing generic identifiability is the half-trek criterion (HTC), which uses the existence of trek systems in the mixed graph to iteratively discover generically invertible linear equation systems in polynomial time. By focusing on edges one at a time, we establish new sufficient and necessary conditions for generic identifiability of edge effects extending those of the HTC. In particular, we show how edge coefficients can be recovered as quotients of subdeterminants of the covariance matrix, which constitutes a determinantal generalization of formulas obtained when using instrumental variables for identification.

Identification in Tree-shaped Linear Structural Causal Models

“…Figure 1: G 1 : the classic instrumental variables (IV) model. G 2 is (generically) identifiable by the TSID algorithm (Weihs et al, 2018) and our method TreeID but for which both the half-trek criterion (HTC) and the ACID algorithm (Kumor et al, 2020) fail. Graph G 3 is identifiable by TreeID but not by TSID.…”

“…A more complex criterionthe criterion for a conditional instrumental variable (cIV) -considers these correlations of V 0 conditionally on another set of variables (Bowden and Turkington, 1984;Pearl, 2001;van der Zander et al, 2015). Other criteria and methods to identify some coefficients in specific graphs involve instrumental sets (IS) (Brito and Pearl, 2002a;Brito, 2010;Brito and Pearl, 2002b;, halftreks (HTC) (Foygel et al, 2012), auxiliary instrumental variables (aIV) (Chen et al, 2015), determinantal instrumental variables (tsIV) which results in the TSID algorithm (Weihs et al, 2018), instrumental cutsets (Kumor et al, 2019), or auxiliary cutsets which result in the ACID algorithm (Kumor et al, 2020).…”

“…Identification of Tree Graphs with the Stateof-the-Art-Methods. The algorithm ACID (Kumor et al, 2020), which subsumes previous stateof-the-art methods, including cIV, IS, aIV, and HTC, as well as the TSID approach (Weihs et al, 2018) belong currently to the most prominent and powerful methods for identification of structural coefficients in linear causal models. They are significantly more efficient compared to the general Gröbner bases approach.…”

“…Several of the tree graphs, like G 1 and G 2 in Fig. 1, can be identified with the TSID algorithm implementing the tsIV criterion (Weihs et al, 2018). This criterion, however, still leaves a significant number of tree graphs unidentified.…”

Structure Learning for Cyclic Linear Causal Models

“…For models with feedback loops and/or latent variables, however, the definition of an appropriate statistical score is non-trivial as the model parameters need not be identifiable and, consequently, the model dimension may differ from the number of parameters that are used to specify the model. Although methods exist to detect identifiability or lack thereof [11,19,33], it is generally unclear when a linear causal model with feedback loops or latent variables is of the dimension expected from a parameter count [4].…”