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A good starting point is chapters 9 & 10 from [0]. Then many of the same topics are re-discussed in the second half of the book through the lens of Bayesian hierarchical models.

Another good reference is [1]. Rubin invented a lot of observational data methods for correcting to measure causal effect. Imbens is also a prolific author in this area, and even just googling for propensity model papers from Imbens will leads to many methods and many other papers.

[0]: < http://www.stat.columbia.edu/~gelman/arm/ >

[1]: < https://www.amazon.com/Causal-Inference-Statistics-Biomedica... >




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