Obviously if you I the sum, I just want to know the die1-die2? The only problem is that the signed difference looks like a uniform distribution with width dependent on the sum—the signed difference can range from 11 possibilities (-5 to 5) down to 1 (0).
So what I think you do is you put all the differences onto the same scale by constructing a “unitless difference,” which will actually be defined as a uniform distribution.
Rather than having the difference be a single number in a chunk of the number line that changes in size, you construct a big set of ordered points of fixed size equal to the least common multiple of the number of possible differences for all sums. If you think of a difference not as a number, but as a uniform distribution on the set of possible differences, then you can just “scale up” this distribution from its set of variable into the big set of constant size, and sample from this distribution to forget the sum but remember the most information about the difference.
Obviously if you I the sum, I just want to know the die1-die2? The only problem is that the signed difference looks like a uniform distribution with width dependent on the sum—the signed difference can range from 11 possibilities (-5 to 5) down to 1 (0).
So what I think you do is you put all the differences onto the same scale by constructing a “unitless difference,” which will actually be defined as a uniform distribution.
Rather than having the difference be a single number in a chunk of the number line that changes in size, you construct a big set of ordered points of fixed size equal to the least common multiple of the number of possible differences for all sums. If you think of a difference not as a number, but as a uniform distribution on the set of possible differences, then you can just “scale up” this distribution from its set of variable into the big set of constant size, and sample from this distribution to forget the sum but remember the most information about the difference.
EDIT: I shouldn’t do math while tired.