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The Mediation Formula: A guide to the assessment of causal pathways in nonlinear models
 STATISTICAL CAUSALITY. FORTHCOMING.
, 2011
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On Identification and Inference for Direct Effects
, 2009
"... Consider the query: Does a binary treatment X have a causal effect on a response Y through a causal pathway that does not involve the intermediate variable M? This query is often rephrased as: Does X have a direct causal effect on Y not through M? Direct effects have been formally defined in three d ..."
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Consider the query: Does a binary treatment X have a causal effect on a response Y through a causal pathway that does not involve the intermediate variable M? This query is often rephrased as: Does X have a direct causal effect on Y not through M? Direct effects have been formally defined in three different ways: the controlled direct effects (CDE), the natural direct effects (i.e. pure and total direct effects PDE and TDE), and the principal stratum direct effects (PSDE). In this issue of the journal, Hafeman and VanderWeele (H&V) 7 provide novel minimal or near minimal conditions for identification of the CDE, PDE and TDE but do not consider the PSDE. In this commentary, we review inference for direct effects and the results of H&V. We also review the close relationship between the direct effects literature and the literature on instrumental variables and Mendelian randomization. 1 Formal Definitions To proceed, we review the formal definitions of the three types of direct effects. We first consider a study with baseline covariates C, a dichotomous treatment X
When two become one: the limits of causality analysis of brain dynamics. PLoS One 2012
"... Biological systems often consist of multiple interacting subsystems, the brain being a prominent example. To understand the functions of such systems it is important to analyze if and how the subsystems interact and to describe the effect of these interactions. In this work we investigate the extent ..."
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Biological systems often consist of multiple interacting subsystems, the brain being a prominent example. To understand the functions of such systems it is important to analyze if and how the subsystems interact and to describe the effect of these interactions. In this work we investigate the extent to which the causeandeffect framework is applicable to such interacting subsystems. We base our work on a standard notion of causal effects and define a new concept called natural causal effect. This new concept takes into account that when studying interactions in biological systems, one is often not interested in the effect of perturbations that alter the dynamics. The interest is instead in how the causal connections participate in the generation of the observed natural dynamics. We identify the constraints on the structure of the causal connections that determine the existence of natural causal effects. In particular, we show that the influence of the causal connections on the natural dynamics of the system often cannot be analyzed in terms of the causal effect of one subsystem on another. Only when the causing subsystem is autonomous with respect to the rest can this interpretation be made. We note that subsystems in the brain are often bidirectionally connected, which means that interactions rarely should be quantified in terms of causeandeffect. We furthermore introduce a framework for how natural causal effects can be characterized when they exist. Our work also has important consequences for the interpretation of other approaches commonly applied to study causality in the brain. Specifically, we discuss how the notion of natural causal effects can be
Analysis of the Binary Instrumental Variable Model
"... We give an explicit geometric characterization of the set of distributions over counterfactuals that are compatible with a given observed joint distribution fortheobservablesinthebinary instrumental variable model. This paper will appear as Chapter 25 in Heuristics, Probability and Causality: A Trib ..."
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We give an explicit geometric characterization of the set of distributions over counterfactuals that are compatible with a given observed joint distribution fortheobservablesinthebinary instrumental variable model. This paper will appear as Chapter 25 in Heuristics, Probability and Causality: A Tribute to Pearlâ€™s seminal work on instrumental variables [Chickering andPearl1996;BalkeandPearl 1997] for discrete data represented a leap forwards in terms of understanding: Pearl showed that, contrary to what many had supposed based on linear models, in the discrete case the assumption that a variable was an instrument could be subjected to empirical test. In
Causal Bounds and Observable Constraints for Nondeterministic Models
"... Conditional independence relations involving latent variables do not necessarily imply observable independences. They may imply inequality constraints on observable parameters and causal bounds, which can be used for falsification and identification. The literature on computing such constraints ofte ..."
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Conditional independence relations involving latent variables do not necessarily imply observable independences. They may imply inequality constraints on observable parameters and causal bounds, which can be used for falsification and identification. The literature on computing such constraints often involve a deterministic underlying data generating process in a counterfactual framework. If an analyst is ignorant of the nature of the underlying mechanisms then they may wish to use a model which allows the underlying mechanisms to be probabilistic. A method of computation for a weaker model without any determinism is given here and demonstrated for the instrumental variable model, though applicable to other models. The approach is based on the analysis of mappings with convex polytopes in a decision theoretic framework and can be implemented in readily available polyhedral computation software. Well known constraints and bounds are replicated in a probabilistic model and novel ones are computed for instrumental variable models without nondeterministic versions of the randomization, exclusion restriction and monotonicity assumptions respectively.
Summary
"... For many applications of machine learning the goal is to predict the value of a vector c given the value of a vector x of input features. In a classification problem c represents a discrete class label, whereas in a regression problem it corresponds to one or more continuous variables. From a probab ..."
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For many applications of machine learning the goal is to predict the value of a vector c given the value of a vector x of input features. In a classification problem c represents a discrete class label, whereas in a regression problem it corresponds to one or more continuous variables. From a probabilistic perspective, the goal is to find the conditional distribution p(cx). The most common approach to this problem is to represent the conditional distribution using a parametric model, and then to determine the parameters using a training set consisting of pairs {xn, cn} of input vectors along with their corresponding target output vectors. The resulting conditional distribution can be used to make predictions of c for new values of x. This is known as a discriminative approach, since the conditional distribution discriminates directly between the different values of c. An alternative approach is to find the joint distribution p(x, c), expressed for instance as a parametric model, and then subsequently uses this joint distribution to evaluate the conditional p(cx) in order to make predictions of c
The Mediation Formula:
, 2011
"... A guide to the assessment of causal pathways in nonlinear models ..."