This is a great question, and actually you might want to invest further giving that you are writing a paper on this (I gather some information from your tweet of what you are trying to do):
-
find_MAPoptimized free parameters on the unbounded scale (in many cases it is also desired as you dont need to specify the boundary of parameters), see a related discussion https://github.com/pymc-devs/pymc3/issues/2545. -
However, the model log_prob that
pm.find_MAP“sees” is different from the onepm.sample“sees”: transforming the bounded free parameters to Real you need to account for the volume changes, however, if you optimized the log_prob you get somewhat unexpected result (there is more discussion here https://github.com/pymc-devs/pymc3/issues/2482)
As a result, we decide that for find_MAP it will optimized the log_prob without the Jacobian correction (https://github.com/pymc-devs/pymc3/issues/2482#issuecomment-320569496) but still on the unbounded space, in this case you get the same as if you are optimizing with the bounded version.
As implication for your paper, I think the you would get the same MLE if you hand written your likelihood function, but you should be aware that the posterior HMC is sampling from is different from the likelihood you are optimizing - it might make your point stronger to argue the importance of using MC samples.