MLE for the fraction of high ratings

Question

Customer reviews are bucketed as low, mid, or high, treated as independent draws with fixed probabilities summing to 1. Out of 300 reviews the counts are 40 low, 110 mid, 150 high.

What is the maximum likelihood estimate of the probability of a high rating?

Accepted Answer

The multinomial MLE, obtained by maximizing \sumj nj \ln pj under the constraint \sumj pj = 1, is the vector of observed proportions \hat pj = nj / n. For high ratings, \hat p{	ext{high}} = \frac{150}{300} = 0.5. Takeaway. Sort the data, count each bucket, and divide by the total: that is the multinomial MLE.

MLE for the fraction of high ratings

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