🚀 go-pugleaf

This is probably a silly question, but I can't learn if I don't ask
questions:

(1) I have M observations x1,x2,x3,....,xM

(2) I have N sets of unknown values U(1),U(2),...,U(N).
    We will refer to the elements of the sets as
    U(1) = { U(1,1) , U(1,2) , ..... , etc }
    U(2) = { U(2,1) , U(2,2) , ..... , etc }
    ... you get the idea.

    Remember: the U(i,j) are unknown .
    

(3) I hypothesize that each of my x(i) is the sum of
    one element chosen from U(1),U(2),.... . That is :

    x(i) = U(1,i1[i]) + U(2,i2[i]) + U(3,i3[i]) + ... + U(N,iN[i])
 
    where at least one of the i1[i],i2[i],i3[i],..,iN[i] is different for
    each of the x(i).

Questions :
(A) How can I get an estimate of the probability that this is true ?
(B) Can I compare the results I get for different N ( see (2) above ) ?
    i.e. - is there a way to determine the best value of N ?

Thankyou in advance.
--David Chasman