Question: From the standpoint of statistician (or a practitioner), can one infer causality using propensity scores with an observational study (not an experiment)?
Please, do not want to start a flame war or a fanatical debate.
Background: Within our stat PhD program, we've only touched on causal inference through working groups and a few topic sessions. However, there are some very prominent researchers in other departments (e.g. HDFS, Sociology) who are actively using them.
I've already witnessed some pretty heated debate on this issue. It is not my intention to start one here. That said, what references have you encountered? What viewpoints do you have? For example, one argument I've heard against propensity scores as a causal inference technique is that one can never infer causality due omitted variable bias -- if you leave out something important, you break the causal chain. Is this an unresolvable problem?
Disclaimer: This question may not have a correct answer -- completely cool with clicking cw, but I'm personally very interested in the responses and would be happy with a few good references which include real-world examples.