Inferring the causal structure that links n observables is usually based upon detecting statistical dependences and choosing simple graphs that make the joint measure Markovian. Here we argue why causal inference is also possible when only single observations are present.
We develop a theory how to generate causal graphs explaining similarities between single objects. To this end, we replace the notion of conditional stochastic independence in the causal Markov condition with the vanishing of conditional algorithmic mutual information and describe the corresponding causal inference rules.
We explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional probability densities, making it possible to select among Markov equivalent causal graphs. This insight provides a theoretical foundation of a heuristic principle proposed in earlier work.
We also discuss how to replace Kolmogorov complexity with decidable complexity criteria. This can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution.
We postulate a principle stating that the initial condition of a physical system is typically algorithmically independent of the dynamical law. We argue that this links thermodynamics and causal inference. On the one hand, it entails behaviour that is similar to the usual arrow of time. On the other hand, it motivates a statistical asymmetry between cause and effect that has recently postulated in the field of causal inference, namely, that the probability distribution P(cause) contains no information about the conditional distribution P(effect|cause) and vice versa, while P(effect) may contai…
ProPublica's analysis of recidivism predictions produced by Correctional Offender Management Profiling for Alternative Sanctions (COMPAS) software tool for the task, has shown that the predictions were racially biased against African American defendants. We analyze the COMPAS data using a causal reformulation of the underlying algorithmic fairness problem. Specifically, we assess whether COMPAS exhibits racial bias against African American defendants using FACT, a recently introduced causality grounded measure of algorithmic fairness. We use the Neyman-Rubin potential outcomes framework for ca…
Our evolution as a species made a huge step forward when we understood the relationships between causes and effects. These associations may be trivial for some events, but they are not in complex scenarios. To rigorously prove that some occurrences are caused by others, causal theory and causal inference were formalized, introducing the $do$-operator and its associated rules. The main goal of this report is to review and implement in Python some algorithms to compute conditional and non-conditional causal queries from observational data. To this end, we first present some basic background know…
Pennsylvania State University（宾夕法尼亚州立大学）；Purdue University（普渡大学）
As virtually all aspects of our lives are increasingly impacted by algorithmic decision making systems, it is incumbent upon us as a society to ensure such systems do not become instruments of unfair discrimination on the basis of gender, race, ethnicity, religion, etc. We consider the problem of determining whether the decisions made by such systems are discriminatory, through the lens of causal models. We introduce two definitions of group fairness grounded in causality: fair on average causal effect (FACE), and fair on average causal effect on the treated (FACT). We use the Rubin-Neyman pot…
This manuscript contributes a general and practical framework for casting a Markov process model of a system at equilibrium as a structural causal model, and carrying out counterfactual inference. Markov processes mathematically describe the mechanisms in the system, and predict the system's equilibrium behavior upon intervention, but do not support counterfactual inference. In contrast, structural causal models support counterfactual inference, but do not identify the mechanisms. This manuscript leverages the benefits of both approaches. We define the structural causal models in terms of the …
Spirtes, Glymour and Scheines [Causation, Prediction, and Search (1993) Springer] described a pointwise consistent estimator of the Markov equivalence class of any causal structure that can be represented by a directed acyclic graph for any parametric family with a uniformly consistent test of conditional independence, under the Causal Markov and Causal Faithfulness assumptions. Robins et al. [Biometrika 90 (2003) 491-515], however, proved that there are no uniformly consistent estimators of Markov equivalence classes of causal structures under those assumptions. Subsequently, Kalisch and Bühl…
Assessing the magnitude of cause-and-effect relations is one of the central challenges found throughout the empirical sciences. The problem of identification of causal effects is concerned with determining whether a causal effect can be computed from a combination of observational data and substantive knowledge about the domain under investigation, which is formally expressed in the form of a causal graph. In many practical settings, however, the knowledge available for the researcher is not strong enough so as to specify a unique causal graph. Another line of investigation attempts to use obs…
University of South Australia（南澳大学）；Lingnan University（岭南大学）；California Institute of Technology（加州理工学院）；Shenzhen University（深圳大学）
In recent years the possibility of relaxing the so-called Faithfulness assumption in automated causal discovery has been investigated. The investigation showed (1) that the Faithfulness assumption can be weakened in various ways that in an important sense preserve its power, and (2) that weakening of Faithfulness may help to speed up methods based on Answer Set Programming. However, this line of work has so far only considered the discovery of causal models without latent variables. In this paper, we study weakenings of Faithfulness for constraint-based discovery of semi-Markovian causal model…
Kalisch and Bühlmann (2007) showed that for linear Gaussian models, under the Causal Markov Assumption, the Strong Causal Faithfulness Assumption, and the assumption of causal sufficiency, the PC algorithm is a uniformly consistent estimator of the Markov Equivalence Class of the true causal DAG for linear Gaussian models; it follows from this that for the identifiable causal effects in the Markov Equivalence Class, there are uniformly consistent estimators of causal effects as well. The $k$-Triangle-Faithfulness Assumption is a strictly weaker assumption that avoids some implausible implicati…
Feature selection is an important problem in machine learning, which aims to select variables that lead to an optimal predictive model. In this paper, we focus on feature selection for post-intervention outcome prediction from pre-intervention variables. We are motivated by healthcare settings, where the goal is often to select the treatment that will maximize a specific patient's outcome; however, we often do not have sufficient randomized control trial data to identify well the conditional treatment effect. We show how we can use observational data to improve feature selection and effect est…
Constraint-based methods are one of the main approaches for causal structure learning that are particularly valued as they are asymptotically guaranteed to find a structure that is Markov equivalent to the causal graph of the system. On the other hand, they may require an exponentially large number of conditional independence (CI) tests in the number of variables of the system. In this paper, we propose a novel recursive constraint-based method for causal structure learning that significantly reduces the required number of CI tests compared to the existing literature. The idea of the proposed …
In observational studies, when a total causal effect of interest is not identified, the set of all possible effects can be reported instead. This typically occurs when the underlying causal DAG is only known up to a Markov equivalence class, or a refinement thereof due to background knowledge. As such, the class of possible causal DAGs is represented by a maximally oriented partially directed acyclic graph (MPDAG), which contains both directed and undirected edges. We characterize the minimal additional edge orientations required to identify a given total effect. A recursive algorithm is then …
We formalize constraint-based structure learning of the "true" causal graph from observed data when unobserved variables are also existent. We provide conditions for a "natural" family of constraint-based structure-learning algorithms that output graphs that are Markov equivalent to the causal graph. Under the faithfulness assumption, this natural family contains all exact structure-learning algorithms. We also provide a set of assumptions, under which any natural structure-learning algorithm outputs Markov equivalent graphs to the causal graph. These assumptions can be thought of as a relaxat…
MIT（麻省理工学院）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM Research（IBM研究院）
Entropic causal inference is a framework for inferring the causal direction between two categorical variables from observational data. The central assumption is that the amount of unobserved randomness in the system is not too large. This unobserved randomness is measured by the entropy of the exogenous variable in the underlying structural causal model, which governs the causal relation between the observed variables. Kocaoglu et al. conjectured that the causal direction is identifiable when the entropy of the exogenous variable is not too large. In this paper, we prove a variant of their con…
Causal structure learning with momentum: Sampling distributions over Markov Equivalence Classes of DAGs
Chalmers University of Technology（查尔姆斯理工大学）；University of Gothenburg（哥德堡大学）；Institute for Theoretical Computer Science（理论计算机科学研究所）；University of Lübeck（吕贝克大学）
Northeastern University（东北大学）；GE Global Research（通用电气全球研究中心）
Recent developments have linked causal inference with Algorithmic Information Theory, and methods have been developed that utilize Conditional Kolmogorov Complexity to determine causation between two random variables. We present a method for inferring causal direction between continuous variables by using an MDL Binning technique for data discretization and complexity calculation. Our method captures the shape of the data and uses it to determine which variable has more information about the other. Its high predictive performance and robustness is shown on several real world use cases.
Harvard University（哈佛大学）；Institute for Quantitative Social Science（定量社会科学研究所）；University of Massachusetts（马萨诸塞大学）
Using the concept of principal stratification from the causal inference literature, we introduce a new notion of fairness, called principal fairness, for human and algorithmic decision-making. The key idea is that one should not discriminate among individuals who would be similarly affected by the decision. Unlike the existing statistical definitions of fairness, principal fairness explicitly accounts for the fact that individuals can be impacted by the decision. Furthermore, we explain how principal fairness differs from the existing causality-based fairness criteria. In contrast to the count…
Stanford University（斯坦福大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Tübingen AI Center（图宾根人工智能中心）
Regulators and academics are increasingly interested in the causal effect that algorithmic actions of a digital platform have on consumption. We introduce a general causal inference problem we call the steerability of consumption that abstracts many settings of interest. Focusing on observational designs and exploiting the structure of the problem, we exhibit a set of assumptions for causal identifiability that significantly weaken the often unrealistic overlap assumptions of standard designs. The key novelty of our approach is to explicitly model the dynamics of consumption over time, viewing…
On the physics of nested Markov models: a generalized probabilistic theory perspective
QICI Quantum Information and Computation Initiative, School of Computing and Data Science, The University of Hong Kong, Hong Kong SAR, China（量子信息与计算倡议，计算与数据科学学院，香港大学，香港特别行政区，中国）；Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China（交叉信息科学研究所，清华大学，北京，中国）；Shanghai Qi Zhi Institute, Shanghai, China（上海启智研究所，上海，中国）；Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada（理论物理研究所，滑铁卢，安大略，加拿大）
The Australian National University（澳大利亚国立大学）；Monash University（莫纳什大学）
Causal discovery automates the learning of causal Bayesian networks from data and has been of active interest from their beginning. With the sourcing of large data sets off the internet, interest in scaling up to very large data sets has grown. One approach to this is to parallelize search using Markov Blanket (MB) discovery as a first step, followed by a process of combining MBs in a global causal model. We develop and explore three new methods of MB discovery using Minimum Message Length (MML) and compare them empirically to the best existing methods, whether developed specifically as MB dis…
In this discussion we consider why it is important to estimate causal effect parameters well even they are not identified, propose a partially identified approach for causal inference in the presence of colliders, point out an under-appreciated advantage of double robustness, discuss the relative difficulty of independence testing versus regression, and finally commend Häggström for her exploration of causal inference with high-dimensional confounding, while making a call for further research in this same vein.
Real causal processes may contain feedback loops and change over time. In this paper, we model cycles and non-stationary distributions using a mixture of directed acyclic graphs (DAGs). We then study the conditional independence (CI) relations induced by a density that factorizes according to a mixture of DAGs in two steps. First, we generalize d-separation for a single DAG to mixture d-separation for a mixture of DAGs. We then utilize the mixture d-separation criterion to derive a global Markov property that allows us to read off the CI relations induced by a mixture of DAGs using a particula…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Radboud University Nijmegen（拉德堡德大学奈梅亨分校）
This work addresses the following question: Under what assumptions on the data generating process can one infer the causal graph from the joint distribution? The approach taken by conditional independence-based causal discovery methods is based on two assumptions: the Markov condition and faithfulness. It has been shown that under these assumptions the causal graph can be identified up to Markov equivalence (some arrows remain undirected) using methods like the PC algorithm. In this work we propose an alternative by defining Identifiable Functional Model Classes (IFMOCs). As our main theorem w…
Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Saarland University（萨尔大学）
We consider the problem of inferring the directed, causal graph from observational data, assuming no hidden confounders. We take an information theoretic approach, and make three main contributions. First, we show how through algorithmic information theory we can obtain SCI, a highly robust, effective and computationally efficient test for conditional independence---and show it outperforms the state of the art when applied in constraint-based inference methods such as stable PC. Second, building upon on SCI, we show how to tell apart the parents and children of a given node based on the algori…
Université Libre de Bruxelles（布鲁塞尔自由大学）；Interuniversity Institute of Bioinformatics in Brussels（布鲁塞尔大学生物信息学研究所）
The relationship between statistical dependency and causality lies at the heart of all statistical approaches to causal inference. Recent results in the ChaLearn cause-effect pair challenge have shown that causal directionality can be inferred with good accuracy also in Markov indistinguishable configurations thanks to data driven approaches. This paper proposes a supervised machine learning approach to infer the existence of a directed causal link between two variables in multivariate settings with $n>2$ variables. The approach relies on the asymmetry of some conditional (in)dependence relati…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Sharif University of Technology（谢里夫理工大学）；Georgia Institute of Technology（佐治亚理工学院）；Carnegie Mellon University（卡内基梅隆大学）
A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure--roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. I…
Amazon Research（亚马逊研究院）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Discussions on causal relations in real life often consider variables for which the definition of causality is unclear since the notion of interventions on the respective variables is obscure. Asking 'what qualifies an action for being an intervention on the variable X' raises the question whether the action impacted all other variables only through X or directly, which implicitly refers to a causal model. To avoid this known circularity, we instead suggest a notion of 'phenomenological causality' whose basic concept is a set of elementary actions. Then the causal structure is defined such tha…
We study the performance of Local Causal Discovery (LCD), a simple and efficient constraint-based method for causal discovery, in predicting causal effects in large-scale gene expression data. We construct practical estimators specific to the high-dimensional regime. Inspired by the ICP algorithm, we use an optional preselection method and two different statistical tests. Empirically, the resulting LCD estimator is seen to closely approach the accuracy of ICP, the state-of-the-art method, while it is algorithmically simpler and computationally more efficient.
We introduce computational causal inference as an interdisciplinary field across causal inference, algorithms design and numerical computing. The field aims to develop software specializing in causal inference that can analyze massive datasets with a variety of causal effects, in a performant, general, and robust way. The focus on software improves research agility, and enables causal inference to be easily integrated into large engineering systems. In particular, we use computational causal inference to deepen the relationship between causal inference, online experimentation, and algorithmic …
We introduce the minimal maximally predictive models (ε-machines) of processes generated by certain hidden semi-Markov models. Their causal states are either hybrid discrete-continuous or continuous random variables and causal-state transitions are described by partial differential equations. Closed-form expressions are given for statistical complexities, excess entropies, and differential information anatomy rates. We present a complete analysis of the ε-machines of continuous-time renewal processes and, then, extend this to processes generated by unifilar hidden semi-Markov models and semi-M…
Markov random fields area popular model for high-dimensional probability distributions. Over the years, many mathematical, statistical and algorithmic problems on them have been studied. Until recently, the only known algorithms for provably learning them relied on exhaustive search, correlation decay or various incoherence assumptions. Bresler gave an algorithm for learning general Ising models on bounded degree graphs. His approach was based on a structural result about mutual information in Ising models. Here we take a more conceptual approach to proving lower bounds on the mutual informati…
We give a category-theoretic treatment of causal models that formalizes the syntax for causal reasoning over a directed acyclic graph (DAG) by associating a free Markov category with the DAG in a canonical way. This framework enables us to define and study important concepts in causal reasoning from an abstract and "purely causal" point of view, such as causal independence/separation, causal conditionals, and decomposition of intervention effects. Our results regarding these concepts abstract away from the details of the commonly adopted causal models such as (recursive) structural equation mo…
Max-linear Bayesian networks have emerged as highly applicable models for causal inference via extreme value data. However, conditional independence (CI) for max-linear Bayesian networks behaves differently than for classical Gaussian Bayesian networks. We establish the parallel between the two theories via tropicalization, and establish the surprising result that the Markov equivalence classes for max-linear Bayesian networks coincide with the ones obtained by regular CI. Our paper opens up many problems at the intersection of extreme value statistics, causal inference and tropical geometry.
