So I did some further simulation to demonstrate my point:
Using the same model, I fixed the b and sd to the real value, and plot the 2D conditional log-likelihood of a and omega:
Immediately you can see, there are a lot of weight around a = 0, but let’s zoom in to the correct value:
Indeed, there is a local maximum around 3.5, but it is not particularly salient.
Moreover, zoom in around omega=0
Again, you see a lot of volume.
All in all, I think it should convince you that why this model is difficult to sample, and why Metropolis although seemly return a better looking chain, the estimation is wrong.
Notebook here: https://github.com/junpenglao/Planet_Sakaar_Data_Science/blob/master/PyMC3QnA/discourse_1190.ipynb