This is a follow-up to this question, which might be too difficult. So I break the problem down. What I need to find is the log-likelihood of the following function:
$$f(x, a_{sg}, a_{pl}, p) = \begin{cases} \frac{1}{1+\log_{n}(a_{sg})} & , x\leq Singulars \\ min[\frac{1}{1+\log_{n}(a_{pl})}, \frac{1}{1+\log_{n}(a_{sg})}+p] & , x\leq Plurals \end{cases}$$
It would also be helpful for me to know how I could approach such a problem. An analytical solution is preferred, but I'd be happy also with an empirical approach, e.g. based on Bayesian sampling or similar.