I am currently exploring methods for maximum likelihood estimation with binned data and have observed that many approaches utilize either the multinomial or Poisson distribution to calculate the likelihood. I was wondering why it is common to use these methods instead of calculating the probability of each bin range directly using the cumulative distribution function (CDF) and then multiplying the probabilities for all data points that fall within each bin.
Is there a specific advantage to using the multinomial or Poisson approach over the this straightforward probability multiplication method?
