I have data set X = {x_1, x_2, \ldots, x_N}, each x_i is a d-dimensional vector, where scalars are from some finite field (In practice they are categories, represented by integers from 1...C).

If my data would be 1-d, then multinomial density would be ok to model it. How do I extend it to multidimensional case? If scalars in vector are iid, then I guess, I could just model it by d-product of multinomial densities? Did I miss something?