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The Rubin Causal Model (RCM), also called Potential Outcome Framework, assumes any unit in a population has potential outcomes under any treatment relevant in a study. For example $Y_1$ denotes the outcome under treatment, $Y_0$ the outcome under control. In a non-randomized experiment the fact that in expectation the quantity

$$E(Y_1|T=1) - E(Y_0|T=0)$$

is observed in expectation as the mean difference between treated and untreated units, where $T$ denotes the treatment assignment, causes selection bias against the average treatment effect

$$E(Y_1-Y_0).$$

The key problem is that part of the data needed for causal inference is unobserved, in particular $E(Y_1|T=0)$ and $E(Y_0|T=1)$. There are several approaches to inference about this treatment effect, prominently weighting, stratification, or some other form of matching on the propensity score.

An alternative approach tries to estimate the unobserved potential outcome distributions $P(Y_1,Y_0)$ directly using Bayesian techniques, such as multiple imputation. What are key papers that attempt causal inference by solving the missing data problem by multiple imputation or other Bayesian techniques?

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    $\begingroup$ Not sure I understand this question. Rubin is a Bayesian and his approach (though stemming from Neyman, who was not) is to put the whole thing in a missing data framework. But the inference framework is distinct from the causal framework. $\endgroup$ – conjugateprior Jul 20 '16 at 21:34
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    $\begingroup$ If you are looking for a Bayesian approach within the potential outcome framework tradition, ch.8 of Imbens and Rubin's 'Causal Inference' textbook might be a place to start. $\endgroup$ – conjugateprior Jul 20 '16 at 21:35
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$$E(Y_1−Y_0)$$ is the quantity we would like to learn about. Counterfactuals per se are not observed, so we need to make further assumptions to write this counterfactual quantity in terms of the observed variables $Y$ and $T$. One way is to assume that $$Y_1, Y_0 \perp T,$$ for example because treatment is randomized. If one further assumes that the observed variable Y obeys $Y = Y_1\cdot T + Y_0 \cdot (1 - T)$, we can write $$E(Y_1−Y_0) = E(Y_1|T = 1) − E(Y_0|T = 0) = E(Y|T = 1) - E(Y|T = 0).$$ The last expression can be estimated in a myriad of ways, because we can actually observe $Y$ and $T$. So the "causal inference"-step itself consists only of justifying and using counterfactual assumptions, and is not directly related to any estimation procedure like the ones you mention. Judea Pearl makes this point very forcefully (e.g., in "Causality", 2009, Cambridge University Press).

To me, multiple imputation of the potential outcome distribution does not make sense. I am also not aware of any paper trying to do this. If one has a sample on $Y$ and $T$, one can make inferences about their population distributions, and if one makes the right assumption (as for example above), these quantities also tell you something about causal effects. An entirely different topic is how to deal with missing values in $Y$ and $T$ (not $Y_t$), and to what extent this is a problem for causal inference. For that, see [1] for a gentle and intuitive, and [2] for a very comprehensive treatment.

[1] Pearl, Judea. "Linear models: A useful “microscope” for causal analysis." Journal of Causal Inference 1.1 (2013): 155-170.

[2] Shpitser, Ilya, Karthika Mohan, and Judea Pearl. Missing data as a causal and probabilistic problem. No. TR-R-454. CALIFORNIA UNIV LOS ANGELES DEPT OF COMPUTER SCIENCE, 2015.

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There's a great paper by Cham & West about this matter, but mostly in the context of propensity scores; you might find some good references in that paper to follow up your pursuit.

Cham, H., & West, S. G. (2016). Propensity Score Analysis With Missing Data. Psychological Methods. http://doi.org/10.1037/met0000076

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