Rating My Professor Using Bayesian Statistics In our lecture, our professor was telling us that "almost any question you might have can be answered using Bayesian Statistics".
Our prof gave the following example: Suppose you have to take a math course next year with some professor, and you would like to know how nice or mean this professor is. You ask 5 students who have taken courses with this professor to rate their experience.

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*Student 1 gives the professor 6/10

*Student 2 gives the professor 0/10

*Student 3 gives the professor 7/10

*Student 4 gives the professor 10/10

*Student 5 gives the professor 1/10

Now suppose you gather some more information on the background of each student:

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*Student 1 seems to be a rational and fair student - although we don't know a lot about Student 1

*Student 2 is very critical of everything and always blames everyone except himself. Student 2 had never taken a math class before.

*Student 3 is a hard working student who tends to get along with everyone

*Student 4 is a very talented student at mathematics and has won many math awards in the past

*Student 5 is also a very talented student at mathematics and has also won many awards in the past

To wrap up the example, the professor mentioned:

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*In the Frequentist example, you would take the average ratings from all such students and conclude that this professor is roughly 5/10

*In the Bayesian example, you would also take the average ratings from all students - but you would try to incorporate "beliefs and credibility" about the opinions from these students, and try to "pool them together" to estimate the rating of this professor.

Our professor then proceeded to make a joke and said that "he is 10/10 no matter what estimation method you use" to which everyone laughed.
But I was wondering - what kind of Bayesian Priors can be used for this example and how would you convert the student opinions into a Prior? The ratings themselves are proportions and thus follow a Binomial Distribution - and in the case of the Binomial Distribution, the Beta Distribution forms a "convenient Prior" (i.e. conjugate). But in this case, how would we decide values of "alpha" and "beta" in this example?
Can someone please comment on this?
Thanks!

You seem to post a lot of questions about your professor, and frankly many of them make me dubious of his authenticity.
Nevertheless, I think the problem suffers from a few misconceptions.

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*First, I disagree that this problem should be treated as binomial.  There are no "trials" to speak of.  However, there are ordinal ratings which suggests that maybe an ordinal model might be better.


*Second, you've given us data and asked how to construct a prior from the data.  An easy way would be to simply fit these data to a model which uses uninformative priors and then pass the posterior as a prior to a new model.  Of course, I don't think that's what you meant.  But since priors are necessarily pre-data, it doesn't make sense to ask "how to convert these [data] into a prior".


*If we agree that a binomial is not the best model, then I think we can satisfactorily answer your question.  In my opinion, a more appropriate model would be an ordinal regression model since the outcomes are discrete and ordered.  We can let the probability of receiving a particular rating be a function of characteristics of the student.
As an example, I suppose you might think that student 2 is more likely to give lower ratings since they are critical and have little experience in math classes.  If we can convert the given information into a covariate, then we can model the ratings as a function of these covariates.  The hard part would be turning these information into a numerical score, but ask a quantitative question and get quantitative answer.  The priors are then placed on the coefficients of the model.  If you suspect that students like student 2 will always yield lower scores, then you could construct your model to force the coefficients for "no experience in math" and "criticality" to only have a specific sign.  That is technically prior information although perhaps it isn't a prior distribution.
