A Survey on Causal Inference <s> INTRODUCTION <s> A discussion of matching, randomization, random sampling, and other methods of controlling extraneous variation is presented. The objective is to specify the benefits of randomization in estimating causal effects of treatments. The basic conclusion is that randomization should be employed whenever possible but that the use of carefully controlled nonrandomized data to estimate causal effects is a reasonable and necessary procedure in many cases. Recent psychological and educational literature has included extensive criticism of the use of nonrandomized studies to estimate causal effects of treatments (e.g., Campbell & Erlebacher, 1970). The implication in much of this literature is that only properly randomized experiments can lead to useful estimates of causal effects. If taken as applying to all fields of study, this position is untenable. Since the extensive use of randomized experiments is limited to the last half century,8 and in fact is not used in much scientific investigation today,4 one is led to the conclusion that most scientific "truths" have been established without using randomized experiments. In addition, most of us successfully determine the causal effects of many of our everyday actions, even interpersonal behaviors, without the benefit of randomization. Even if the position that causal effects of treatments can only be well established from randomized experiments is taken as applying only to the social sciences in which <s> BIB001 </s> A Survey on Causal Inference <s> INTRODUCTION <s> In the portion of the paper translated here, Neyman introduces a model for the analysis of field experiments conducted for the purpose of comparing a number of crop varieties, which makes use of a double-indexed array of unknown potential yields, one index corresponding to varieties and the other to plots. The yield corresponding to only one variety will be observed on any given plot, but through an urn model embodying sampling without replacement from this doubly indexed array, Neyman obtains a formula for the variance of the difference between the averages of the observed yields of two varieties. This variance involves the variance over all plots of the potential yields and the correlation coefficient r between the potential yields of the two varieties on the same plot. Since it is impossible to estimate r directly, Neyman advises taking r = 1, observing that in practice this may lead to using too large an estimated standard deviation, when comparing two variety means. <s> BIB002 </s> A Survey on Causal Inference <s> INTRODUCTION <s> SUMMARY The primary aim of this paper is to show how graphical models can be used as a mathematical language for integrating statistical and subject-matter information. In particular, the paper develops a principled, nonparametric framework for causal inference, in which diagrams are queried to determine if the assumptions available are sufficient for identifying causal effects from nonexperimental data. If so the diagrams can be queried to produce mathematical expressions for causal effects in terms of observed distributions; otherwise, the diagrams can be queried to suggest additional observations or auxiliary experiments from which the desired inferences can be obtained. <s> BIB003 </s> A Survey on Causal Inference <s> INTRODUCTION <s> Abstract: This review presents empirical researchers with recent advances in causal inference, and stresses the paradigmatic shifts that must be undertaken in moving from traditional statistical analysis to causal analysis of multivariate data. Special emphasis is placed on the assumptions that underly all causal inferences, the languages used in formulating those assumptions, the conditional nature of all causal and counterfactual claims, and the methods that have been developed for the assessment of such claims. These advances are illustrated using a general theory of causation based on the Structural Causal Model (SCM) described in Pearl (2000a), which subsumes and uniﬁes other approaches to causation, and provides a coherent mathematical foundation for the analysis of causes and counterfactuals. In particular, the paper surveys the development of mathematical tools for inferring (from a combination of data and assumptions) answers to three types of causal queries: (1) queries about the eﬀects of potential interventions, (also called “causal eﬀects” or “policy evaluation”) (2) queries about probabilities of counterfactuals, (including assessment of “regret,” “attribution” or “causes of eﬀects”) and (3) queries about direct and indirect eﬀects (also known as “mediation”). Finally, the paper deﬁnes the formal and conceptual relationships between the structural and potential-outcome frameworks and presents tools for a symbiotic analysis that uses the strong features of both. <s> BIB004 </s> A Survey on Causal Inference <s> INTRODUCTION <s> When estimating causal effects using observational data, it is desirable to replicate a randomized experiment as closely as possible by obtaining treated and control groups with similar covariate distributions. This goal can often be achieved by choosing well-matched samples of the original treated and control groups, thereby reducing bias due to the covariates. Since the 1970’s, work on matching methods has examined how to best choose treated and control subjects for comparison. Matching methods are gaining popularity in fields such as economics, epidemiology, medicine, and political science. However, until now the literature and related advice has been scattered across disciplines. Researchers who are interested in using matching methods–or developing methods related to matching–do not have a single place to turn to learn about past and current research. This paper provides a structure for thinking about matching methods and guidance on their use, coalescing the existing research (both old and new) and providing a summary of where the literature on matching methods is now and where it should be headed. <s> BIB005 </s> A Survey on Causal Inference <s> INTRODUCTION <s> 1. Introduction to probabilities, graphs, and causal models 2. A theory of inferred causation 3. Causal diagrams and the identification of causal effects 4. Actions, plans, and direct effects 5. Causality and structural models in the social sciences 6. Simpson's paradox, confounding, and collapsibility 7. Structural and counterfactual models 8. Imperfect experiments: bounds and counterfactuals 9. Probability of causation: interpretation and identification Epilogue: the art and science of cause and effect. <s> BIB006 </s> A Survey on Causal Inference <s> INTRODUCTION <s> Correlation implies association, but not causation. Conversely, causation implies association, but not correlation. <s> BIB007 </s> A Survey on Causal Inference <s> INTRODUCTION <s> The era of big data provides researchers with convenient access to copious data. However, people often have little knowledge about it. The increasing prevalence of big data is challenging the traditional methods of learning causality because they are developed for the cases with limited amount of data and solid prior causal knowledge. This survey aims to close the gap between big data and learning causality with a comprehensive and structured review of traditional and frontier methods and a discussion about some open problems of learning causality. We begin with preliminaries of learning causality. Then we categorize and revisit methods of learning causality for the typical problems and data types. After that, we discuss the connections between learning causality and machine learning. At the end, some open problems are presented to show the great potential of learning causality with data. <s> BIB008 </s> A Survey on Causal Inference <s> INTRODUCTION <s> Estimating what would be an individual's potential response to varying levels of exposure to a treatment is of high practical relevance for several important fields, such as healthcare, economics and public policy. However, existing methods for learning to estimate counterfactual outcomes from observational data are either focused on estimating average dose-response curves, or limited to settings with only two treatments that do not have an associated dosage parameter. Here, we present a novel machine-learning approach towards learning counterfactual representations for estimating individual dose-response curves for any number of treatments with continuous dosage parameters with neural networks. Building on the established potential outcomes framework, we introduce performance metrics, model selection criteria, model architectures, and open benchmarks for estimating individual dose-response curves. Our experiments show that the methods developed in this work set a new state-of-the-art in estimating individual dose-response. <s> BIB009
In everyday language, correlation and causality are commonly used interchangeably, although they have quite different interpretations. Correlation indicates a general relationship: two variables are correlated when they display an increasing or decreasing trend BIB007 . Causality is also referred to as cause and effect where the cause is partly responsible for the effect, and the effect is partly dependent on the cause. Causal inference is the process of drawing a conclusion about a causal connection based on the conditions of the occurrence of an effect. The main difference between causal inference and inference of correlation is that the former analyzes the response of the effect variable when the cause is changed BIB004  . It is well known that "correlation does not imply causation. " For example, a study showed that girls have breakfast normally have lightweight than the girls who don't, and thus concluded that having breakfast can help to lose weight. But in fact, these two events may just have correlation instead of causality. Maybe the girls who have breakfast everyday have a better lifestyle, such as exercise frequently, sleep regularly, and have a healthy diet, which finally makes them have lightweight. In this case, having a better lifestyle is the common cause of both having breakfast and lightweight, and thus we also can treat it as a confounder of the causality between having breakfast and lightweight. In many cases, it seems obvious that one action can cause another; however, there exists also many cases that we cannot easily tease out and make sure the relationship. Therefore, learning causality is one dauntingly challenging problem. The most effective way of inferring causality is to conduct a randomized controlled trial, which randomly assigns participants into a treatment group or a control group. As the randomized study is conducted, the only expected difference between the control and treatment groups is the outcome variable being studied. However, in reality, randomized controlled trials are always time-consuming and expensive, and thus the study cannot involve many subjects, which may be not representative of the real-world population a treatment/intervention would eventually target. Another issue is that the randomized controlled trials only focus on the average of samples, and it doesn't explain the mechanism or pertain for individual subjects. In addition, ethical issues also need to be considered in most of the randomized controlled trials, which largely limits its applications. Therefore, instead of the randomized controlled trials, the observational data is a tempting shortcut. Observational data is obtained by the researcher simply observing the subjects without any interfering. That means, the researchers have no control over treatments and subjects, and they just observe the subjects and record data based on their observations. From the observational data, we can find their actions, outcomes, and information about what has occurred, but cannot figure out the mechanism why they took a specific action. For the observational data, the core question is how to get the counterfactual outcome. For example, we want to answer this question "would this patient have different results if he received a different medication?" Answering such counterfactual questions is challenging due to two reasons BIB009 : the first one is that we only observe the factual outcome and never the counterfactual outcomes that would potentially have happened if they have chosen a different treatment option. The second one is that treatments are typically not assigned at random in observational data, which may lead the treated population differs significantly from the general population. To solve these problems in causal inference from observational data, researchers develop various frameworks, including the potential outcome framework BIB001 BIB002 and the structural causal model BIB003 92,  . The potential outcome framework is also known as the Neyman-Rubin Potential Outcomes or the Rubin Causal Model. In the example we mentioned above, a girl would have a particular weight if she had breakfast normally everyday, whereas she would have a different weight if she didn't have breakfast normally. To measure the causal effect of having breakfast normally for a girl, we need to compare the outcomes for the same person under both situations. Obviously, it is impossible to see both potential outcomes at the same time, and one of the potential outcomes is always missing. The potential outcome framework aims to estimate such potential outcomes and then calculate the treatment effect. Therefore, the treatment effect estimation is one of the central problems in causal inference under the potential outcome framework. Another influential framework in causal inference is the structural causal model (SCM), which includes the causal graph and the structural equations. The structural causal model describes the causal mechanisms of a system where a set of variables and the causal relationship among them are modeled by a set of simultaneous structural equations. Causal inference has a close relationship with the machine learning area. In recent years, the magnificent bloom of the machine learning area enhances the development of the causal inference area. Powerful machine learning methods such as decision tree, ensemble methods, deep neural network, are applied to estimate the potential outcome more accurately. In addition to the amelioration on the outcome estimation model, machine learning methods also provide a new aspect to handle the confounders. Benefit from the recently deep representation learning methods, such as generative adversarial neural network, the confounder variables are adjusted by learning the balanced representation for all covariates, so that conditioning on the learned representation, the treatment assignment is independent of the confounder variables. In machine learning, the more data the better. However, in causal inference, the more data alone is not yet enough. Having more data only helps to get more precise estimates, but it cannot make sure these estimates are correct and unbiased. Machine learning methods enhance the development of causal inference, meanwhile, causal inference also helps machine learning methods. The simple pursuit of predictive accuracy is insufficient for modern machine learning research, and correctness and interpretability are also the targets of machine learning methods. Causal inference is starting to help to improve machine learning, such as recommender systems or reinforcement learning. In this article, we provide a comprehensive review of the causal inference methods under the potential outcome framework. We first introduce the basic concepts of the potential outcome framework as well as its three critical assumptions to identify the causal effect. After that, various causal inference methods with these three assumptions are discussed in detail, including re-weighting methods, stratification methods, matching based methods, tree-based methods, representation-based methods, multi-task learning based methods, and meta-learning methods. Additionally, causal effect estimation methods that relax the three assumptions are also described to fulfill the needs in different settings. After introducing various causal effect estimation methods, the real-world applications that the discussed methods have great potential to benefit are discussed, including the advertisement area, recommendation area, medicine area, and reinforcement learning area as the representative examples. To the best of our knowledge, this is the first paper that provides a comprehensive survey for causal inference methods under the potential outcome framework. There also exist several surveys that discuss one category of the causal effect estimation methods, such as the survey of matching based methods BIB005 , survey of tree-based and ensemble-based method  , and the review of dynamic treatment regimes  . For the structural causal model, it is suggested to refer the survey BIB004 or the book BIB006 . We will also briefly discuss the relation and difference between the two causality frameworks at the end of our survey. There is also a survey about learning causality from observational data BIB008 whose content ranges from inferring the causal graph from observational data, structural causal model, potential outcome framework and their connection to machine learning. Compared with the surveys mentioned above, this survey paper mainly focuses on the theoretical background of the potential outcome framework, the representative methods across the statistic domain and machine learning domain, and how this framework and the machine learning area enhance each other. To summarize, our contributions of this survey are as follows: • New taxonomy: We separate various causal inference methods into two major categories based on whether they require the three assumptions of the potential outcome framework. The category requiring three assumptions are further divided into seven sub-categories based on the way to handle the confounder variables. • Comprehensive review: We provide a comprehensive survey of the causal inference methods under the potential outcome framework. In each category, the detailed descriptions of the representative methods, the connection and comparison between the mentioned methods, and the general summation are provided. • Abundant resources: In this survey, we list the state-of-art methods, the benchmark data sets, open-source codes, and representative applications. The rest of the paper is organized as follows. In section 2, the background of the potential outcome framework is introduced, including the basic definitions, the assumptions, and the fundamental problems with their general solutions. In Section 3, the methods under three assumptions are presented. Then, in Section 4, we discuss the problem when some assumptions are not satisfied, and describe the methods that relax those assumptions. Next, we provide experimental guidelines in Section 5. Afterward, in Section 6, the typical applications of causal inference are illustrated. Finally, Section 7 summarizes the paper.