I admit I'm relatively new to propensity scores and causal analysis.
One thing that's not obvious to me as a newcomer is how the "balancing" using propensity scores is mathematically different from what happens when we add covariates in a regression? What's different about the operation, and why is it (or is it) better than adding subpopulation covariates in a regression?
I've seen some studies that do an empirical comparison of the methods, but I haven't seen a good discussion relating the mathematical properties of the two methods and why PSM lends itself to causal interpretations while including regression covariates does not. There also seems to be a lot of confusion and controversy in this field, which makes things even more difficult to pick up.
Any thoughts on this or any pointers to good resources/papers to better understand the distinction? (I'm slowly making my way through Judea Pearl's causality book, so no need to point me to that)