A Survey on Causal Inference <s> Definitions <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> Definitions <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> Definitions <s> Observational studies are rising in importance due to the widespread accumulation of data in fields such as healthcare, education, employment and ecology. We consider the task of answering counterfactual questions such as, "Would this patient have lower blood sugar had she received a different medication?". We propose a new algorithmic framework for counterfactual inference which brings together ideas from domain adaptation and representation learning. In addition to a theoretical justification, we perform an empirical comparison with previous approaches to causal inference from observational data. Our deep learning algorithm significantly outperforms the previous state-of-the-art. <s> BIB003 </s> A Survey on Causal Inference <s> Definitions <s> There is intense interest in applying machine learning to problems of causal inference in fields such as healthcare, economics and education. In particular, individual-level causal inference has important applications such as precision medicine. We give a new theoretical analysis and family of algorithms for predicting individual treatment effect (ITE) from observational data, under the assumption known as strong ignorability. The algorithms learn a "balanced" representation such that the induced treated and control distributions look similar. We give a novel, simple and intuitive generalization-error bound showing that the expected ITE estimation error of a representation is bounded by a sum of the standard generalization-error of that representation and the distance between the treated and control distributions induced by the representation. We use Integral Probability Metrics to measure distances between distributions, deriving explicit bounds for the Wasserstein and Maximum Mean Discrepancy (MMD) distances. Experiments on real and simulated data show the new algorithms match or outperform the state-of-the-art. <s> BIB004
Here we define the notations under the potential outcome framework BIB001 BIB002 , which is logically equivalent to another framework, the structural causal model framework [62] . The foundation of potential outcome framework is that the causality is tied to treatment (or action, manipulation, intervention), applied to a unit  . The treatment effect is obtained by comparing units' potential outcomes of treatments. In the following, we first introduce three essential concepts in causal inference: unit, treatment, and outcome. Definition 1. Unit. A unit is the atomic research object in the treatment effect study. A unit can be a physical object, a firm, a patient, an individual person, or a collection of objects or persons, such as a classroom or a market, at a particular time point  . Under the potential outcome framework, the atomic research objects at different time points are different units. One unit in the dataset is a sample of the whole population, so in this survey, the term "sample" and "unit" are used interchangeably. Definition 2. Treatment. Treatment refers to the action that applies (exposes, or subjects) to a unit. Let W (W ∈ {0, 1, 2, . . . , N W }) denote the treatment, where N W + 1 is the total number of possible treatments. In the aforementioned medicine example, Medicine A is a treatment. Most of the literatures consider the binary treatment, and in this case, the group of units applied with treatment W = 1 is the treated group, and the group of units with W = 0 is the control group. Definition 3. Potential outcome. For each unit-treatment pair, the outcome of that treatment when applied on that unit is the potential outcome  . The potential outcome of treatment with value w is denoted as Y (W = w). Definition 4. Observed outcome. The observed outcome is the outcome of the treatment that is actually applied. The observed outcome is also called factual outcome, and we use Y F to denote it where F stands for "factual". The relation between the potential outcome and the observed outcome is: where w is the treatment actually applied. Definition 5. Counterfactual outcome: Counterfactual outcome is the outcome if the unit had taken another treatment. The counterfactual outcomes are the potential outcomes of the treatments except the one actually taken by the unit. Since a unit can only take one treatment, only one potential outcome can be observed, and the remaining unobserved potential outcomes are the counterfactual outcome. In the multiple treatment case, let Y C F (W = w ′ ) denote the counterfactual outcome of treatment with value w ′ . In the binary treatment case, for notation simplicity, we use Y C F to denote the counterfactual outcome, and Y C F = Y (W = 1 − w), where w is the treatment actually taken by the unit. In the observational data, besides the chosen treatments and the observed outcome, the units' other information is also recorded, and they can be separated as pre-treatment variables and the post-treatment variables. Definition 6. Pre-treatment variables: Pre-treatment variables are the variables that will not be affected by the treatment. Pre-treatment variables are also named as background variables, and they can be patients' demographics, medical history, and etc. Let X denote the pre-treatment variables. Definition 7. post-treatment variables: The post-treatment variables are the variables that are affected by the treatment. One example of post-treatment variables is the intermediate outcome, such as the lab test after taking the medicine in the aforementioned medicine example. In the following sections, the terminology variable refers to the pre-treatment variable unless otherwise specified. Treatment Effect. After introducing the observational data and the key terminologies, the treatment effect can be quantitatively defined using the above definitions. The treatment effect can be measured at the population, treated group, subgroup, and individual levels. To make these definitions clear, here we define the treatment effect under binary treatment, and it can be extended to multiple treatments by comparing their the potential outcomes. At the population level, the treatment effect is named as the Average Treatment Effect (ATE), which is defined as: where Y(W = 1) and Y(W = 0) are the potential treated and control outcome of the whole population respectively. For the treated group, the treatment effect is named as Average Treatment effect on the Treated group (ATT), and it is defined as: where Y(W = 1)|W = 1 and Y(W = 0)|W = 1 are the potential treated and control outcome of the treated group respectively. At the subgroup level, the treatment effect is called Conditional Average Treatment Effect (CATE), which is defined as: where Y(W = 1)|X = x and Y(W = 0)|X = x are the potential treated and control outcome of the subgroup with X = x, respectively. CATE is a common treatment effect measurement under the case where the treatment effect varies across different subgroups, which is also known as the heterogeneous treatment effect. At the individual level, the treatment effect is called Individual Treatment Effect (ITE), and the ITE of unit i is defined as: where Y i (W = 1) and Y i (W = 0) are the potential treated and control outcome of unit i respectively. In some literatures BIB003 BIB004 , the ITE is viewed equivalent to the CATE. Objective. For causal inference, our objective is to estimate the treatment effects from the observational data. Formally speaking, given the observational dataset, where N is the total number of units in the datasets, the goal of the causal inference task is to estimate the treatment effects defined above.