Dear Jessegrabowski,
Thank you for your reply. Nevertheless, it is still unclear to me S-Morten’s decision for this problem at hand.
Since, in his/her case, the data is mostly countable (Poisson distributed), and the Poisson distribution assumes an Exponential linking function for the linear terms, I see no reason for applying an InverseLogit function for his/her likelihood function. As you well described, the constraints of the problem at hand should be strictly positive, from \mathbb{R} → (0, +inf). Therefore, by applying the InverseLogit function, the problem will be constrained to \mathbb{R} → (0, 1), a subset of the original broader Domain of \mathbb{R}.
With respect to the parameter \lambda, depending on the implementation of each statistical package, it is common practice to usually define the inverse of the linking function to the likelihood function (in order to facilitate the integration steps); this is the reason for this original question. Regardless, I see now that we must actually provide the linking function to the Likelihood-function, here in PYMC. If I got it wrong here, please let me know.