Hadn’t seen the dice example, is it from Jaynes? (I don’t yet see why you’re better off randomising)

Well, one way to forget the sum is to generate random pairs of dice for each possible sum and replace one of them with your actual pair. For example, if your dice came up (3 5), you can rewrite your memory with something like “the result was one of (1 1) (2 1) (3 1) (4 1) (4 2) (2 5) (3 5) (4 5) (6 4) (6 5) (6 6)”. Is there a simpler way?

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.

Note that the agent should rewrite its memory with a distribution, not just a list of tuples—e.g. {(1 1): ^{1}⁄_{36}, (2 1): ^{2}⁄_{36}, …}. That way the “posterior” distribution on the sum will match the prior distribution on the sum.

That said, this is basically correct. It matches the answer(s) I got, and is more elegant.

Yeah. I guess I was assuming that the agent knows the list of tuples and also knows that they came from the procedure I described; the distribution follows from that :-)

Well, one way to forget the sum is to generate random pairs of dice for each possible sum and replace one of them with your actual pair. For example, if your dice came up (3 5), you can rewrite your memory with something like “the result was one of (1 1) (2 1) (3 1) (4 1) (4 2) (2 5) (3 5) (4 5) (6 4) (6 5) (6 6)”. Is there a simpler way?

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.

Note that the agent should rewrite its memory with a distribution, not just a list of tuples—e.g. {(1 1):

^{1}⁄_{36}, (2 1):^{2}⁄_{36}, …}. That way the “posterior” distribution on the sum will match the prior distribution on the sum.That said, this is basically correct. It matches the answer(s) I got, and is more elegant.

Yeah. I guess I was assuming that the agent knows the list of tuples and also knows that they came from the procedure I described; the distribution follows from that :-)