Lets say I have movie ratings from different users for multiple films. I can find the beta distribution that best fits all the ratings. I can also find the beta distribution that best fits the ratings for a particular movie. What I would like is to be able to say: "If I pull a rating of a movie out of a hat that is a 3, but I don't know what the movie is or who the user is, what is the most likely distribution of their ratings of movie X?".
Meaning that because people who have rated one movie lower are likely to rate movies lower generally, I should generally get a distribution of probable ratings for movie X with a lower mean.
I can take a movie and get distributions for that movie given a user has rated a movie a 1, a 2 a 3 etc. But this would mean that close but not exact ratings would not support neighbouring distributions and continuous ratings would not work at all.
What would be best is a surface plot probability mass function where the y axis is the given rating of an unknown film, where if I took a slice at that y, I would get a Beta distribution where x is ratings and z is probability.
Does such a distribution exist and how would I best compute the most likely parameters for the distribution?