The programme of the forthcoming UK Causal Inference Meeting "Causal Inference in Health, Economic and Social Sciences" is just out. The short conference will be at the end of the month (28th and 29th of April) at the University of Cambridge.
I indirectly feature as Aidan (who's part of our RDD team) is giving a presentation in one of the sessions. His talk is entitled "The Effect of Prior Beliefs on Causal Effect Estimators within a Bayesian Regression Discontinuity Design" and basically comes out as a follow up to our paper.
The idea is to implement the RDD within a full Bayesian framework and to actually assess carefully what's the gain of doing it this way. As is often the case, in some situations, there may be unwanted impact of the prior on the causal estimators, although generally there are advantages (both computationally and in terms of including extra sources of information $-$ that is crucial, especially when you want to go beyond statistical analysis, in my opinion).