We propose a novel method for sensitivity analysis to unobserved confounding in causal inference. The method builds on a copula-based causal graphical normalizing flow that we term ρ-GNF, where ρ is the sensitivity parameter. The parameter represents the non-causal association between exposure and outcome due to unobserved confounding, which is modeled as a Gaussian copula. In other words, the ρ-GNF enables scholars to estimate the average causal effect (ACE) as a function of ρ, accounting for various confounding strengths. The output of the ρ-GNF is what we term the ρ-curve, which provides the bounds for the ACE given an interval of assumed ρ values. The ρ-curve  also enables scholars to identify the confounding strength required to nullify the ACE. We also propose a Bayesian version of our sensitivity analysis method. Assuming a prior over the sensitivity parameter ρ enables us to derive the posterior distribution over the ACE, which enables us to derive credible intervals. Finally, leveraging on experiments from simulated and real-world data, we show the benefits of our sensitivity analysis method.

Recent research has highlighted the properties that deep-learning inspired causal models such as Deep-Structural Causal Model (Deep-SCM), Causal Autoregressive Flow (CAREFL) and Causal-Graphical Normalizing Flow (c-GNF) should exhibit to guarantee observational and interventional distribution equivalence with the true underlying causal data generating process (DGP), making them suitable for estimating average causal effect (ACE) or conditional ACE (CACE). However, for accurate individual-level causal effect (ICE) estimation and personalized treatment/public-policy formulation, it is crucial to ensure counterfactual equivalence between these models and the DGP. Firstly, we demonstrate that c-GNFs provide counterfactual equivalence under certain monotonicity assumption of the DGP, enabling precise ICE estimation and personalized treatment/public-policy analysis. Secondly, using this counterfactual equivalence of c-GNFs, we perform a counterfactual analysis and personalized public-policy analysis of the impact of International Monetary Fund (IMF) programs on child poverty using large-scale real-world observational data. Our results indicate a reduction in child poverty due to the IMF program at different personalization granularities. Our study also performs sensitivity analyses to assess potential threats to the unconfoundedness assumption and estimates ACE bounds and the E-value. This illustrates the potential of c-GNFs for causal and counterfactual inference in fields such as social, natural, and medical sciences.

We propose a novel sensitivity analysis to unobserved confounding in observational studies using copulas and normalizing flows. Using the idea of interventional equivalence of structural causal models, we develop ρρ-GNF (ρρ-graphical normalizing flow), where ρ∈[−1,+1]ρ∈[−1,+1] is a bounded sensitivity parameter. This parameter represents the back-door non-causal association due to unobserved confounding, and which is encoded with a Gaussian copula. In other words, the ρρ-GNF enables scholars to estimate the average causal effect (ACE) as a function of ρρ, while accounting for various assumed strengths of the unobserved confounding. The output of the ρρ-GNF is what we denote as the ρcurveρcurve that provides the bounds for the ACE given an interval of assumed ρρ values. In particular, the ρcurveρcurve enables scholars to identify the confounding strength required to nullify the ACE, similar to other sensitivity analysis methods (e.g., the E-value). Leveraging on experiments from simulated and real-world data, we show the benefits of ρρ-GNF. One benefit is that the ρρ-GNF uses a Gaussian copula to encode the distribution of the unobserved causes, which is commonly used in many applied settings. This distributional assumption produces narrower ACE bounds compared to other popular sensitivity analysis methods.

We extend path analysis by showing that, for a singly-connected path diagram, the partial covariance of two random variables factorizes over the nodes and edges in the path between the variables. This result allows us to determine the contribution of each node and edge to the partial covariance. It also allows us to show that Simpson's paradox cannot occur in singly-connected path diagrams.

Structural Equation/Causal Models (SEMs/SCMs) are widely used in epidemiology and social sciences to identify and analyze the average causal effect (ACE) and conditional ACE (CACE). Traditional causal effect estimation methods such as Inverse Probability Weighting (IPW) and more recently Regression-With-Residuals (RWR) are widely used - as they avoid the challenging task of identifying the SCM parameters - to estimate ACE and CACE. However, much work remains before traditional estimation methods can be used for counterfactual inference, and for the benefit of Personalized Public Policy Analysis (P3A) in the social sciences. While doctors rely on personalized medicine to tailor treatments to patients in laboratory settings (relatively closed systems), P3A draws inspiration from such tailoring but adapts it for open social systems. In this article, we develop a method for counterfactual inference that we name causal-Graphical Normalizing Flow (c-GNF), facilitating P3A. A major advantage of c-GNF is that it suits the open system in which P3A is conducted. First, we show how c-GNF captures the underlying SCM without making any assumption about functional forms. This capturing capability is enabled by the deep neural networks that model the underlying SCM via observational data likelihood maximization using gradient descent. Second, we propose a novel dequantization trick to deal with discrete variables, which is a limitation of normalizing flows in general. Third, we demonstrate in experiments that c-GNF performs on-par with IPW and RWR in terms of bias and variance for estimating the ACE, when the true functional forms are known, and better when they are unknown. Fourth and most importantly, we conduct counterfactual inference with c-GNFs, demonstrating promising empirical performance. Because IPW and RWR, like other traditional methods, lack the capability of counterfactual inference, c-GNFs will likely play a major role in tailoring personalized treatment, facilitating P3A, optimizing social interventions - in contrast to the current `one-size-fits-all' approach of existing methods.

Acyclic directed mixed graphs (ADMGs) are the graphs used by Pearl (Causality: models, reasoning, and inference. Cambridge University Press, Cambridge, 2009) for causal effect identification. Recently, alternative acyclic directed mixed graphs (aADMGs) have been proposed by Peña (Proceedings of the 32nd conference on uncertainty in artificial intelligence, 577–586, 2016) for causal effect identification in domains with additive noise. Since the ADMG and the aADMG of the domain at hand may encode different model assumptions, it may be that the causal effect of interest is identifiable in one but not in the other. Causal effect identification in ADMGs is well understood. In this paper, we introduce a sound algorithm for identifying arbitrary causal effects from aADMGs. We show that the algorithm follows from a calculus similar to Pearl’s do-calculus. Then, we turn our attention to Andersson–Madigan–Perlman chain graphs, which are a subclass of aADMGs, and propose a factorization for the positive discrete probability distributions that are Markovian with respect to these chain graphs. We also develop an algorithm to perform maximum likelihood estimation of the factors in the factorization.

We introduce a new class of graphical models that generalizes Lauritzen-Wermuth-Frydenbergchain graphs by relaxing the semi-directed acyclity constraint so that only directed cycles are forbidden. Moreover, up to two edges are allowed between any pair of nodes. Specifically, we present local, pairwise and global Markov properties for the new graphical models and prove their equivalence. We also present an equivalent factorization property.

Andersson-Madigan-Perlman chain graphs were originally introduced to represent independence models. They have recently been shown to be suitable for representing causal models with additive noise. In this paper, we present an algorithm for learning causal chain graphs. The algorithm builds on the ideas by \citet{Hoyeretal.2009}, i.e. it exploits the nonlinearities in the data to identify the direction of the causal relationships. We also report experimental results on real-world data.