Identification and Estimation of Joint Probabilities of Potential Outcomes in Observational Studies with Covariate Information

1 Introduction . 1.1 Practical Background . In practical science , it is crucial to evaluate the likelihood of one event causing another event . For example , epidemiologists pay attention to determining the likelihood of a particular exposure being the cause of a particular disease . In order to assess such likelihood from statistical data , the probabilities of causation have been developed , which can be divided into “ necessary causation ” , “ sufficient causation ” , and “ necessary-and-sufficient causation ” . In addition , recently , in the field of explainable artificial intelligence ( XAI ) , it has been pointed out that “ necessity ” , “ sufficiency ” , and “ necessity and suffciency ” are the important concepts of successful explanation and the probabilities of causation play an important role 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . in evaluating these concepts from the probabilistic aspects [ 10 , 22 , 35 ] . For the last decades , the probabilities of causation received increasing attention in the fields of epidemiology , risk analysis , legal reasoning , artificial intelligence , and policy analysis [ e.g. , 4–8 , 10 , 11 , 22 , 26 , 29 , 33 , 35 ] . 1.2 “ Necessity ” , “ Sufficiency ” , and “ Necessity and Sufficiency ” . Necessary cause is often used in the tort-law liability . When a 60-year-old man who worked in an asbestos textile plant develops lung cancer , we are interested in the probability that asbestos had caused it . This quantity to be estimated has been commonly referred to as “ attributable fraction ” or “ assigned share ” [ 4 , 19 ] . It measures how necessary the exposure is for the development of the disease . Pearl [ 24 ] defined it as “ Probability of Necessity ( PN ) ” , which means “ the probability that disease would not have occurred in the absence of exposure , given that disease and exposure did in fact occur ” . The estimation of PN is important to provide a scientific basis for compensation in tort suits . Sufficient cause is often used in health risk assessment . In practice , it is important for public policy makers to predict how a health risk factor would endanger a healthy population [ 15 , 16 , 21 ] . The measure for it has been referred to as “ susceptibility ” , which measures how sufficient a risk factor is to make an individual develop a disease . Pearl [ 24 ] defined the measure of such prediction as “ Probability of Sufficiency ( PS ) ” , which means “ the probability that a healthy unexposed individual would have developed the disease had he been exposed ” . The assessment of PS helps public policy makers to make or revise regulations in order to minimize the occurrence of diseases . With the definitions of PN and PS , it would be a natural extension to inquire for the “ Probability of Necessity and Sufficiency ( PNS ) ” , which shows how likely an individual is to be affected by both ways [ 24 ] . PNS measures the probability that an exposure is a necessary and sufficient cause for a disease , that is , an exposure is the actual cause of a disease . Then , different from previous works where “ probability of causation ” is only referred to PN [ e.g. , 4 , 7 ] , Pearl [ 24 ] defined three types of “ probability of causation ” , PN , PS , and PNS . Pearl [ 26 ] and Tian and Pearl [ 33 ] developed formal semantics for the probabilities of causation based on the structural models of counterfactuals . They presented the formal definitions of the probability of necessity ( PN ) , the probability of sufficiency ( PS ) , and the probability of necessity and sufficiency ( PNS ) . These probabilities of causation are formulated based on the joint probabilities of two potential outcomes . Since one can not simultaneously observe the results of the same subjects exposed and unexposed in reality , even successful randomized experiments have failed to identify these quantities [ 26 , pp.284–285 ] . Tian and Pearl [ 33 ] addressed the problem by showing how to bound these quantities using data obtained from experimental and observational studies . Their bounds are sharp under minimal assumptions concerning the data generating process . Kuroki and Cai [ 17 ] derived narrower bounds of the probabilities of causation than Tian-Pearl ’ s bounds using covariate information . Recently , Galhotra et al . [ 10 ] generalized Tian-Pearl ’ s bounds from binary to arbitrary variables in the context of explainable artificial intelligence ( XAI ) . However , it has been often pointed out that these existing bounds are too wide to evaluate the probabilities of causation . To solve the problem , Tian and Pearl [ 33 ] and Galhotra et al . [ 10 ] also stated that the probabilities of causation are identifiable if the monotonicity can be assumed and the causal risks are identifiable and Pearl [ 26 ] showed that the specific functional relationships between cause and effect lead to the identification of the probabilities of causation . 1.3 Contribution . The aim of this paper is to discuss the identification and estimation problems of the joint probabilities of potential outcomes , which provides a general class including the probabilities of causation . The joint probabilities of potential outcomes are the building-blocks of causal inference , because the causal risk is also identifiable if they are identifiable , but not vise versa . In the context of the natural direct and indirect effects [ 25 ] , under the assumption of no unmeasured confounding , Robins and Richardson [ 30 ] stated that the joint probabilities of potential outcomes are identifiable if ( i ) two potential outcomes are independent or ( ii ) one potential outcome is deterministically formulated as the function of the other potential outcome . In addition , in the context of unit selection problems , Li and Pearl [ 20 ] showed that the linear combination of the joint probabilities of potential outcomes is identifiable under the gain equality . These existing researches show that the evaluation of the joint probabilities of potential outcomes play an important role in solving various problems of causal inference . Although considerable efforts have been devoted toward establishing the identifiability criteria and solving the confounding problem of causal risks , there has been limited discussion on how to identify the joint probabilities of potential outcomes when the present assumptions are violated . To solve the problem , this paper proposes two types of novel identification conditions for the joint probabilities of potential outcomes without existing assumptions — one uses both causal risks and one covariate ( Theorem 1 ) and the other uses two covariates without causal risks ( Theorem 2 ) . The results of this paper also provide a counterexample to “ sensitivity to the generative process ” [ 26 , pp . 284–285 ] , in the sense that it is not required to specify the functional relationships that connect a cause and an effect . In addition , unlike the effect restoration proposed by Kuroki and Pearl [ 18 ] , the assumption of the given number of categories of an unmeasured confounder is not required . Furthermore , from Theorem 2 , it is remarkable that there are some situations where the causal risks are identifiable even if the existing identification conditions do not hold . When the joint probabilities of potential outcomes are identifiable through the proposed conditions , the estimation problem of the joint probabilities of potential outcomes reduces to that of singular models and thus they can not be evaluated by standard statistical estimation methods . To solve the problem , this paper proposes a new statistical method based on the augmented Lagrangian method [ 3 ] and shows the asymptotic normality of the proposed estimators . Given space constraints , the proofs , the details on the statistical estimation method , some numerical experiments , and the case study are provided in the supplementary material . 2 Preliminary . This section introduces the potential outcomes used to discuss our problems . For simplicity , we consider the case where an exposure variable X ∈ { x1 , x0 } ( x1 is exposed ; x0 is unexposed ) and an outcome variable Y ∈ { y1 , y0 } ( y1 is disease ; y0 is non-disease ) are dichotomous . In addition , let p ( X = x , Y = y ) = p ( x , y ) , p ( Y = y |X = x ) = p ( y |x ) , and p ( X = x ) = p ( x ) be the joint probability of ( X , Y ) = ( x , y ) , the conditional probability of Y = y given X = x , and the marginal probability of X = x for x ∈ { x1 , x0 } and y ∈ { y1 , y0 } , respectively . Similar notation is used for other probabilities . Here , in this paper , we will assume that readers are familiar with the basic theory of causal inference [ 13 , 26 ] . Especially , for the graph-theoretic terminology and the basic theory of structural causal models used in this paper , we refer the readers to Pearl [ 26 ] . In principle , for x ∈ { x1 , x0 } , the i-th of the N subjects has a potential outcome Yx ( i ) that would have resulted if X had been x , denoted as X ( i ) = x . Here , note that the subject ensures a deterministic relationship between two variables X and Y in the semantics of structural causal models [ 26 ] . In addition , the present paper assumes the stable unit treatment value assumption , which can be summarized as follows : ( i ) the exposure status of any subject does not affect the outcomes of the other subjects ( no interference ) and ( ii ) the exposures of all subjects are comparable ( no variation in exposure ) . Thus , when the N subjects in the study are considered as random samples from the population of interest , X ( i ) and Yx ( i ) are referred to as the values of random variables X and Yx respectively , and thus the causal risk of X = x on Y = y is defined as p ( Yx = y ) . Similar notation is used for other potential outcomes . For the i-th subject , the potential outcome Yx is observed only if X is x . This property is called the consistency [ 13 , 26 , 28 ] , which is formulated as X ( i ) = x=⇒Yx ( i ) = Y ( i ) . When a randomized experiment is conducted and compliance is perfect , since X is independent of ( Yx0 , Yx1 ) , the causal risk is identifiable and is given by p ( Yx = y ) = p ( y |x ) . Here , “ identifiable ” means that the causal quantities , such as p ( Yx = y ) , can be estimated consistently from a joint probability of observed variables . In contrast , when it is difficult to conduct a randomized experiment and only observational data are available , we can evaluate the causal risk according to the conditionally-ignorable-treatment-assignment condition [ 31 ] , or graphically , the back-door criterion [ 26 ] . In other words , for the exposure X , if there exists such a set S of observed covariates that X is conditionally independent of ( Yx0 , Yx1 ) given S , we can say that treatment assignment is conditionally ignorable given S. Then , the causal effects are estimable using S as p ( Yx = y ) = Es [ p ( y |x , S ) ] . Here , Es [ p ( y |x , S ) ] is the expectation of p ( y |x , S ) regarding S. Although there are other identification conditions that can be used to solve our problem [ e.g. , 26 , 34 ] , this paper does not cover them due to space constraints . Following Pearl [ 26 ] , we define three probabilities of causation , that is , the probability of necessity ( PN ) , the probability of sufficiency ( PS ) , and the probability of necessity and sufficiency ( PNS ) . PN stands for the probability p ( Yx0 = y0 |X = x1 , Y = y1 ) that a disease would not have occurred in the absence of an exposure , given that the exposure and the disease occurred . PS stands for the probability p ( Yx1 = y1 |X = x0 , Y = y0 ) that a healthy unexposed subject would have contracted the disease upon exposure . PNS stands for the probability p ( Yx1 = y1 , Yx0 = y0 ) that an exposure is a necessary and sufficient cause for a disease . Since PN , PS , and PNS involve the joint probabilities of potential outcomes , even under successful randomized experiments , they can not be estimated from the observed data without additional assumptions [ 26 ] .

Summary: This paper proposes two nonstandard identification strategies for the joint distribution of counterfactuals. One is based on a single proxy $Z$ for the unknown confounder with at least $4$ levels, such that $p(y|do(z))$ is identifiable. The other one is based two proxies $Z$ and $W$ with at least $4$ levels for each. Both identification strategies are fully nonparametric. They are given by solutions of a complex system which are nonlinear in the observed distributions. Beyond identification, the paper also proposes estimation procedures and proves the asymptotic normality of the estimators. The estimation procedures are essentially the plug-in version of the system involved in the identification strategy.