first of all sorry about the title. I couldn't think anything better.
Let me describe the problem first and ask later. Imagine I have 1500 observations of the results of two dice being played (I don't know how many times each die was played). E.g.:
Sample1 1 187 2 168 3 164 4 320* 5 187 6 174
Seeing that there is something odd with these dice (Number 4 seems to be over-represented), imagine I'm able physically to measure the result bias from each die, E.g.
Dice 1(fair) Dice 2(unfair)  0.1666667 0.04  0.1666667 0.04  0.1666667 0.04  0.1666667 0.80*  0.1666667 0.04  0.1666667 0.04
My goal is to measure the proportion of the 1500 observations each die contributed to. One way of doing this is by adding a parameter P (proportion) and perform a non-negative least square regression. However this is limited since it does not consider the observed results a distribution (right?). Another solution may be to use Maximum likelihood estimation. And here is the question. What is the likelihood function that would model this problem? p(O|b,P). Where b is the bias signature and P are the proportion of throws from each die.
The idea here is to have a generic solution. Let's say I have more dice, 10 for example, each on with a specific bias (signature). Is it possible to recover how many times each die was played?