Assume a simple example motivating a causal research design. Say that I collect a data set on rural counties in Texas and I wish to understand if rainfall causes a change in crop sales. Working with this observational data, I run a regression, conditioning on a necessary adjustment set (to the best of my capability) and run a sensitivity analysis to examine how the point estimate shifts when exposed to varying levels of hypothetical unobserved confounding.
Under such a scenario, I will still estimate information "quantifying uncertainty" (confidence intervals, credible intervals, etc.). However, if:
- 1: Identifying assumptions ensure that, with satisfaction of these assumptions, an estimate is causal and
- 2: Under selection on observables, I can quantify uncertainty of the satisfaction of identifying assumptions
then what information are statistical inferential tools providing? It seems to me that P(H) is determined by the satisfaction of identifying assumptions, not by any sort of Bayesian or frequentist quantification of uncertainty. If I wish to evaluate the uncertainty of P(H), does not a sensitivity analysis of identifying assumptions do this very thing?
I can see the value of quantifying the uncertainty of P(D) given that the data used to estimate a causal effect may not be representative of the broader population and may be insufficient to approximate the causal parameter that exists in the broader population. However, this caveat does not seem to satisfy my understanding of how statistical inference interacts with causal inference when either:
- 1: The data is the population itself (so P(D) may not be of interest anymore).
- 2: One pursues a Bayesian approach where one is interested in P(H|D) rather than P(D|$H_0$). Again, does not the satisfaction of identifying assumptions and sensitivity analyses of these assumptions address P(H)?
I recognize that these topics are not competing and that there is simply confusion on my end for how these goals interact. I appreciate any feedback and suggested readings on the topic.