Alan Jacobs and I have been working on figuring out principles for simultaneously drawing inferences from qualitative and quantitative data.
The key idea is that when scholars use qualitative inference they update beliefs about causal effects (or more, generally about their model of the world, ) by making inferences using data about many facts of a given case (). They estimate a posterior . Quantitative scholars update beliefs about causal effects by making inferences using data about a few facts about many cases (), forming posterior . From there it's not such a huge thing to make integrated inferences of the form . Simple as that sounds, people do not do this, but doing it opens up many insights about how we learn from cases and how we aggregate knowledge.
APSR paper on inferences from qualitative and quantitative data here
New: Working paper on dags, theory, and qualitative inference from causal models here
Video Presentation (CU Feb 2016) here
Rmd Replication of Natural Resources illustration here