Structurally Determined Inequality Constraints on Correlations in the Cycle of Linear Dependencies
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CYBERNETICS STRUCTURALLY DETERMINED INEQUALITY CONSTRAINTS ON CORRELATIONS IN THE CYCLE OF LINEAR DEPENDENCIES
O. S. Balabanov
UDC 004.855:519.216
Abstract. Several simple inequality constraints on correlations in a rhombus-like causal model (structured as a cycle with one collider) are formulated and proved. These constraints follow from the linearity and Markov properties of the model. The presented inequalities are specific to the basic model and are incorrect for alternative models that differ in Markov properties due to the presence of an additional edge (connection). The plausibility of the violation of these inequalities in alternative models is evaluated by stochastic simulation. It is shown that the presented inequalities are useful for model verification under partial observability. Keywords: correlation, inequality constraint, system of linear structural equations, rhombus-like structure, Markov property.
It is well known that the structure of a system of dependencies implies definite constraints on the relation of dependence indices in a model. Markov properties of a causal network are expressed as equalities for conditional probabilities [1, 2]. If not all variables of a model are observed, then it is impossible to verify the corresponding Markov properties. In such a situation, for testing the model, one has to resort to testing alternative constraints and criteria that can be computed from probability distributions of smaller dimensions. General algebraic methods for deriving the constraints implied by the structure of a causal network are presented in [3]. The Verma constraint [4] on distributions of conditional probabilities is well known to specialists in causal models. This constraint characterizes a model of an instrumental variable in which there is a “mediator” variable between cause and effect, and this mediator is associated with a confounder through the cause. New results for discrete models can be found in [5]. The equalities derived in [6] characterize linear models with several hidden variables. These equalities are established for the sum of terms with different signs. For practice, it is important that the constraints established for a model allow one to construct a testing criterion with a wide critical region. In this sense, simple (but nontrivial) constraints are more promising. The most well-known result is the inequality for correlations that is established by the J. S. Bell theorem (for a definite quantum system) [7]. The majority of pragmatic criteria are developed for treelike structures of models and linear forms of dependencies (or for binary variables). In the practice of data analysis, tetrad constraints (equalities of paired products of correlations) are widely used that act in linear models in which no less than four variables are connected by a common “cause” [1, 8]. Constraints on relations of correlations for some nonlinear (monotone) forms of dependencies are also well-known [9]. If a linear model contains only three indicator variables, then “tetrad” is inoperati
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