Graphical methods for inequality constraints in marginalized DAGs

We present a graphical approach to deriving inequality constraints for directed acyclic graph (DAG) models, where some variables are unobserved. In particular we show that the observed distribution of a discrete model is always restricted if any two observed variables are neither adjacent in the graph, nor share a latent parent; this generalizes the well known instrumental inequality. The method also provides inequalities on interventional distributions, which can be used to bound causal effects. All these constraints are characterized in terms of a new graphical separation criterion, providing an easy and intuitive method for their derivation.Comment: A final version will appear in the proceedings of the 22nd Workshop on Machine Learning and Signal Processing, 201

Про характерні співвідношення кореляцій в деяких системах лінійних структуральних рівнянь

Для ймовірнісної лінійної моделі з циклічною структурою із чотирма змінними знайдено і доведено два простих обмеження типу нерівність на наборі кореляцій. Кожне з цих обмежень (що включає дві та три кореляції відповідно) дає змогу спростувати базову модель на користь альтернативної моделі, яка відрізняється додатковим “діагональним” зв'язком.Для вероятностной линейной модели с цикличной структурой с четырьмя переменными найдены и доказаны два простых ограничения типа неравенства на наборе корреляций. Каждое из этих ограничений (включающее две и три корреляции соответственно) даёт возможность опровергнуть базовую модель в пользу альтернативной модели, которая отличается дополнительной “диагональной” связью.For a probabilistic linear model of cyclic structure with four variables, we prove two simple inequality-type constraints on the set of correlations. Each of the inequalities (comprising two and three correlations, respectively) facilitates the rejection of the basic model in favor of an alternative model, which differs in that it contains an additional “diagonal” connection

On the causal interpretation of acyclic mixed graphs under multivariate normality

In multivariate statistics, acyclic mixed graphs with directed and bidirected edges are widely used for compact representation of dependence structures that can arise in the presence of hidden (i.e., latent or unobserved) variables. Indeed, under multivariate normality, every mixed graph corresponds to a set of covariance matrices that contains as a full-dimensional subset the covariance matrices associated with a causally interpretable acyclic digraph. This digraph generally has some of its nodes corresponding to hidden variables. We seek to clarify for which mixed graphs there exists an acyclic digraph whose hidden variable model coincides with the mixed graph model. Restricting to the tractable setting of chain graphs and multivariate normality, we show that decomposability of the bidirected part of the chain graph is necessary and sufficient for equality between the mixed graph model and some hidden variable model given by an acyclic digraph

Model testing for causal models

Finding cause-effect relationships is the central aim of many studies in the physical, behavioral, social and biological sciences. We consider two well-known mathematical causal models: Structural equation models and causal Bayesian networks. When we hypothesize a causal model, that model often imposes constraints on the statistics of the data collected. These constraints enable us to test or falsify the hypothesized causal model. The goal of our research is to develop efficient and reliable methods to test a causal model or distinguish between causal models using various types of constraints. For linear structural equation models, we investigate the problem of generating a small number of constraints in the form of zero partial correlations, providing an efficient way to test hypothesized models. We study linear structural equation models with correlated errors focusing on the graphical aspects of the models. We provide a set of local Markov properties and prove that they are equivalent to the global Markov property. For causal Bayesian networks, we study equality and inequality constraints imposed on data and investigate a way to use these constraints for model testing and selection. For equality constraints, we formulate an implicitization problem and show how we may reduce the complexity of the problem. We also study the algebraic structure of the equality constraints. For inequality constraints, we present a class of inequality constraints on both nonexperimental and interventional distributions