Hi @Bob_Carpenter, many thanks for the very detailed comments. Sorry for getting back with delay.
I believe that I have indeed a situation where I have two measurements / measurement models with the same underlying process.
I said competing when trying to express that the two data sets are not equivalent even when projecting to the same scale, so I want to combine information from both ends and where in conflict, interpret that as samples from a noisy process and let the model capture higher uncertainty where relevant.
Indeed, it seems like the beta regression gives me more uncertain parameter information. I followed your advice and modelled the two data sets seperately, with the results confirming your hypothesis as far as I can see. The figures below show 90% CIs (whisker length) and densities of the Poisson/Beta parameters’ posterior distribution, roughly projected to the same scale (in line with the model code in the OP). The data sets have exactly the same number of observations per wave.
Also thanks for the other comments & hints, I considered them carefully, in some cases it was the intended behavior, in others I reparameterised the model. However, the weakness of beta regression phaenomena remained unchanged.
When assuming the Beta regression to be the root of the problem (and considering that I had very similar observations when using an Exponential distribution initially, and Gamma too):
- What causes the weakness? Is that a general feature of the Beta distribution, or introduced by the way I parameterise it, or specific to the data fitted?
- In case I had no other choice but parameterising the model that way, how would I, in view of the figures above, model a joint latent parameter that gives more weight to the Beta information such that it would contribute more, compensating for the regression weakness?
I sense that a gap in my theoretical knowledge prevent me from fully understanding what I get to observe here, I’d be thankful if you could give me the cue for some further reading. Same goes for modelling techniques that I’m likely not even aware of, making me being stuck. Thanks a lot!
(By the way, as a solution to just the case at hand, I found a workaround that could have been obvious to me all the time: multiplying the ratios with the assumed distribution location of the normalisation factor i.e. the assumed (mean of) total daily count, and passing their integer version to a Poisson too. Works in this special case, results below. I’m still interested in how I could have come there without dropping the Beta distr.)
results in:
