Imagine an a la carte buffet with n different rooms. On entering the buffet you pick a room (let's say American, Mexican or Italian food) where you stay for the duration of your visit.
Once in a room, you have a full selection of small dishes to pick from and most people choose multiples of several different dishes, e.g. 1 burrito, 2 salad, 4 tacos, 2 churros. Presumably these are neither independent nor random choices, but you can't get into the heads of your customers. Also, although the rooms are themed (and people are more likely to order according to the theme they've chosen) every dish is available in every room.
How would you model the distribution of dishes in a visit to a room, given complete data?
How could you quantify the deviation from expectation of a given visitor's selection?
Because the dish selection isn't necessarily independent I think I would do the first according to a Dirichlet process, but I'm not sure that's right.
