Hi Frank!
Sorry for being so out-of-sync with the course, I just managed to complete the 5th session which was a rather math-heavy topic for a non-statistician, I guess that’s why the discussion here hasn’t exactly been brewing 
Anyway, I would like to ask for some clarification regarding the prior assumptions in these calculations.
In the part “Decoding the Effect of the Prior” you do address this to some extent, but I think I would need a little more elaboration to properly understand it. What I can decode from your examples:
(1) Flat priors with infinite standard deviations don’t impose much constraints on the data (not surprising)
(2) Only having very constrained prior distributions (small SD) would make large posterior effect sizes very unlikely (and thus increase the required n by a lot), but more reasonable assumptions actually don’t add a whole lot of extra n.
From a clinician’s standpoint, making a reasonable estimation of μ0 (eg a treatment effect) and σ0 (a believable interval of potential values) seems achievable in the right context (see the valuable discussion here). However what I would feel much more uncertain about is the shape of the distribution itself. I understand that when using frequentist methods one assumes these implicitly, without having to specify it case-by-case. However, with the Bayesian examples of BBR, assumptions such as switching to the gamma distribution and then specifying the α and β for this distribution seems arbitrary. Could one make a similar “sensitivity analysis” for the choice of prior distribution and its additional characteristics? Are there maybe some rules of thumb, for choosing appropriate distributions (and parameters) for certain types of variables in clinical contexts (eg. time-to-event, ordinal scales, etc)?
In the 2019 O’Hagan paper on elicitation of expert opinion, he talks about fitting distribution curves to the three estimates elicited but doesn’t give much guidance as to how one could avoid overfitting. Any other relevant sources I should read on this, primarily from a non-statisticians perspective?
Thank you again!
Aron