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We propose an empirical Bayes formulation of the structure learning problem, where the prior specification assumes that all node variables have the same error variance, an assumption known to ensure the identifiability of the underlying causal directed acyclic graph. To facilitate efficient posterior computation, we approximate the posterior probability of each ordering by that of a best directed acyclic graph model, which naturally leads to an order-based Markov chain Monte Carlo algorithm. Strong selection consistency for our model in high-dimensional settings is proved under a condition that allows heterogeneous error variances, and the mixing behaviour of our sampler is theoretically investigated. Furthermore, we propose a new iterative top-down algorithm, which quickly yields an approximate solution to the structure learning problem and can be used to initialize the Markov chain Monte Carlo sampler. We demonstrate that our method outperforms other state-of-the-art algorithms under various simulation settings, and conclude the paper with a single-cell real-data study illustrating practical advantages of the proposed method. [ABSTRACT FROM AUTHOR]

Directed acyclic graphs provide an effective framework for learning causal relationships among variables given multivariate observations. Under pure observational data, directed acyclic graphs encoding the same conditional independencies cannot be distinguished and are collected into Markov equivalence classes. In many contexts, however, observational measurements are supplemented by interventional data that improve directed acyclic graph identifiability and enhance causal effect estimation. We propose a Bayesian framework for multivariate data partially generated after stochastic interventions. To this end, we introduce an effective prior elicitation procedure leading to a closed-form expression for the directed acyclic graph marginal likelihood and guaranteeing score equivalence among directed acyclic graphs that are Markov equivalent post intervention. Under the Gaussian setting, we show, in terms of posterior ratio consistency, that the true network will be asymptotically recovered, regardless of the specific distribution of the intervened variables and of the relative asymptotic dominance between observational and interventional measurements. We validate our theoretical results via simulation and we implement a Markov chain Monte Carlo sampler for posterior inference on the space of directed acyclic graphs on both synthetic and biological protein expression data. [ABSTRACT FROM AUTHOR]

We study efficient estimation of an interventional mean associated with a point exposure treatment under a causal graphical model represented by a directed acyclic graph without hidden variables. Under such a model, a subset of the variables may be uninformative, in that failure to measure them neither precludes identification of the interventional mean nor changes the semiparametric variance bound for regular estimators of it. We develop a set of graphical criteria that are sound and complete for eliminating all the uninformative variables, so that the cost of measuring them can be saved without sacrificing estimation efficiency, which could be useful when designing a planned observational or randomized study. Further, we construct a reduced directed acyclic graph on the set of informative variables only. We show that the interventional mean is identified from the marginal law by the g-formula (Robins, 1986) associated with the reduced graph, and the semiparametric variance bounds for estimating the interventional mean under the original and the reduced graphical model agree. The g-formula is an irreducible, efficient identifying formula in the sense that the nonparametric estimator of the formula, under regularity conditions, is asymptotically efficient under the original causal graphical model, and no formula with this property exists that depends only on a strict subset of the variables. [ABSTRACT FROM AUTHOR]

Drawing causal inference with observational studies is the central pillar of many disciplines. One sufficient condition for identifying the causal effect is that the treatment-outcome relationship is unconfounded conditional on the observed covariates. It is often believed that the more covariates we condition on, the more plausible this unconfoundedness assumption is. This belief has had a huge impact on practical causal inference, suggesting that we should adjust for all pretreatment covariates. However, when there is unmeasured confounding between the treatment and outcome, estimators adjusting for some pretreatment covariate might have greater bias than estimators without adjusting for this covariate. This kind of covariate is called a bias amplifier, and includes instrumental variables that are independent of the confounder, and affect the outcome only through the treatment. Previously, theoretical results for this phenomenon have been established only for linear models. We fill in this gap in the literature by providing a general theory, showing that this phenomenon happens under a wide class of models satisfying certain monotonicity assumptions. We further show that when the treatment follows an additive or multiplicative model conditional on the instrumental variable and the confounder, these monotonicity assumptions can be interpreted as the signs of the arrows of the causal diagrams.

We propose a linear-time, single-pass, top-down algorithm for multiple testing on directed acyclic graphs, where nodes represent hypotheses and edges specify a partial ordering in which the hypotheses must be tested. The procedure is guaranteed to reject a sub-directed acyclic graph with bounded false discovery rate while satisfying the logical constraint that a rejected node's parents must also be rejected. It is designed for sequential testing settings where the directed acyclic graph structure is known a priori but the |$p$| -values are obtained selectively, such as in a sequence of experiments; however, the algorithm is also applicable in nonsequential settings where all |$p$| -values can be calculated in advance, such as in model selection. Our algorithm provably controls the false discovery rate under independence, positive dependence or arbitrary dependence of the |$p$| -values and specializes to known algorithms in the special cases of trees and line graphs; it simplifies to the classical Benjamini–Hochberg procedure when the directed acyclic graph has no edges. We explore the empirical performance of our algorithm through simulations and analysis of a real dataset corresponding to a gene ontology, and we demonstrate its favourable performance in terms of computational time and power. [ABSTRACT FROM AUTHOR]

Multivariate geostatistics is based on modelling all covariances between all possible combinations of two or more variables at any sets of locations in a continuously indexed domain. Multivariate spatial covariance models need to be built with care, since any covariance matrix that is derived from such a model must be nonnegative-definite. In this article, we develop a conditional approach for spatial-model construction whose validity conditions are easy to check. We start with bivariate spatial covariance models and go on to demonstrate the approach's connection to multivariate models defined by networks of spatial variables. In some circumstances, such as modelling respiratory illness conditional on air pollution, the direction of conditional dependence is clear. When it is not, the two directional models can be compared. More generally, the graph structure of the network reduces the number of possible models to compare. Model selection then amounts to finding possible causative links in the network. We demonstrate our conditional approach on surface temperature and pressure data, where the role of the two variables is seen to be asymmetric. [ABSTRACT FROM AUTHOR]

We consider the problem of multiple hypothesis testing when there is a logical nested structure to the hypotheses. When one hypothesis is nested inside another, the outer hypothesis must be false if the inner hypothesis is false. We model the nested structure as a directed acyclic graph, including chain and tree graphs as special cases. Each node in the graph is a hypothesis and rejecting a node requires also rejecting all of its ancestors. We propose a general framework for adjusting node-level test statistics using the known logical constraints. Within this framework, we study a smoothing procedure that combines each node with all of its descendants to form a more powerful statistic. We prove a broad class of smoothing strategies can be used with existing selection procedures to control the familywise error rate, false discovery exceedance rate, or false discovery rate, so long as the original test statistics are independent under the null. When the null statistics are not independent but are derived from positively-correlated normal observations, we prove control for all three error rates when the smoothing method is arithmetic averaging of the observations. Simulations and an application to a real biology dataset demonstrate that smoothing leads to substantial power gains.

We give a definition of a bounded edge within the causal directed acyclic graph framework. A bounded edge generalizes the notion of a signed edge and is defined in terms of bounds on a ratio of survivor probabilities. We derive rules concerning the propagation of bounds. Bounds on causal effects in the presence of unmeasured confounding are also derived using bounds related to specific edges on a graph. We illustrate the theory developed by an example concerning estimating the effect of antihistamine treatment on asthma in the presence of unmeasured confounding. [ABSTRACT FROM PUBLISHER]

Directed acyclic graphs are commonly used to represent causal relationships among random variables in graphical models. Applications of these models arise in the study of physical and biological systems where directed edges between nodes represent the influence of components of the system on each other. Estimation of directed graphs from observational data is computationally NP-hard. In addition, directed graphs with the same structure may be indistinguishable based on observations alone. When the nodes exhibit a natural ordering, the problem of estimating directed graphs reduces to the problem of estimating the structure of the network. In this paper, we propose an efficient penalized likelihood method for estimation of the adjacency matrix of directed acyclic graphs, when variables inherit a natural ordering. We study variable selection consistency of lasso and adaptive lasso penalties in high-dimensional sparse settings, and propose an error-based choice for selecting the tuning parameter. We show that although the lasso is only variable selection consistent under stringent conditions, the adaptive lasso can consistently estimate the true graph under the usual regularity assumptions. [ABSTRACT FROM PUBLISHER]

We consider variable selection in high-dimensional linear models where the number of covariates greatly exceeds the sample size. We introduce the new concept of partial faithfulness and use it to infer associations between the covariates and the response. Under partial faithfulness, we develop a simplified version of the pc algorithm (Spirtes et al., 2000), which is computationally feasible even with thousands of covariates and provides consistent variable selection under conditions on the random design matrix that are of a different nature than coherence conditions for penalty-based approaches like the lasso. Simulations and application to real data show that our method is competitive compared to penalty-based approaches. We provide an efficient implementation of the algorithm in the R-package pcalg. [ABSTRACT FROM PUBLISHER]

This paper considers the problem of using observational data in the presence of selection bias to identify causal effects in the framework of linear structural equation models. We propose a criterion for testing whether or not observed statistical dependencies among variables are generated by conditioning on a common response variable. When the answer is affirmative, we further provide formulations for recovering the covariance matrix of the whole population from that of the selected population. The results of this paper provide guidance for reliable causal inference, based on the recovered covariance matrix obtained from the statistical information with selection bias. [ABSTRACT FROM AUTHOR]

We study criteria for identifiability of path analysis models with one hidden variable. We first derive sufficient criteria for identification of models in which marginalisation is carried out over the hidden variable. The sufficient criteria are based on the structure of the directed acyclic graph associated with the path analysis model and can be derived from the graph. We treat further the identification of models when the hidden variable is conditioned on and establish connections with the extended skew-normal distribution. Finally it is shown that the derived conditions extend the existing graphical criteria for identification. [ABSTRACT FROM AUTHOR]