MLE not knowing exactly the sample data

Question

I am learning statistics for Machine Learning and have to answer to this basic question from the following extract: "Imagine a machine learning class where the probability that a student gets an 'A' grade is P(A) = 1/2, a 'B' grade P(B) = , a 'C' grade P(C) = , and a 'D' grade P(D) = . We are told that 14 students get a 'C' and 28 students get a `D'. Our goal is to obtain a maximum likelihood estimate of .

b) Assume now we don't know how many students got exactly an 'A' or exactly a 'B'. But we do know that 49 students got either an A or B. Therefore, a and b are unknown values where a + b = 49. Write the function to be optimized under the MLE."

I think I managed to calculate the first MLE for the given sample, but I do not quite understand how can I do it without knowing the values of 'a' and 'b' and cannot seem to figure out how to solve it using Bayes' Theorem.

Answer 1

The full likelihood, based on the complete data, is $$ L(\mu)=(\frac12)^a \mu^b (3\mu)^c (\frac12-4\mu)^d $$ But you only know $a+b$, then we make a likelihood using that $P(a ~\text{or}~ b)= \frac12+\mu$, giving the observed-data likelihood $$ L^*=(\frac12+\mu)^{a+b} (3\mu)^c (\frac12-4\mu)^d. $$