When reverse engineering the spinner for differential privacy and trying to get a privacy budget of 1.1, why devote 50% of the spinner to yes and cut down the 'no', instead of cutting the same proportion out of each?
Great question! On the slide Reverse Engineering the Spinner (1) we need two probabilities: the probability of observing YES when the true state is YES, and the probability of observing YES when the true state is NO. I think what you are asking is whether the proportion that goes to NO is arbitrary. It is. I just have to divide the space between YES and TRUTH so that the correct ratio is achieved. Let's say I have three areas: AT, AY, and AN for TRUTH, YES, and NO, respectively. Like on the slide I will work out that Pr[YES|YES] = AT + AY and Pr[YES|NO] = AY. The ratio is thus (AT+AY)/AY = 1.1, and AT = (0.1)AY. That's the same as the slide so far. Now I can plug in any value I want (within reason, obviously) for AY, and get the corresponding value for AT and AN. Or I can go the other way. Suppose I devote 0.25 of the spinner to NO. Then I end up with approximately 0.682 for YES and 0.0682 for NO. Hope this helps!
When reverse engineering the spinner for differential privacy and trying to get a privacy budget of 1.1, why devote 50% of the spinner to yes and cut down the 'no', instead of cutting the same proportion out of each?
Great question! On the slide Reverse Engineering the Spinner (1) we need two probabilities: the probability of observing YES when the true state is YES, and the probability of observing YES when the true state is NO.
I think what you are asking is whether the proportion that goes to NO is arbitrary. It is. I just have to divide the space between YES and TRUTH so that the correct ratio is achieved.
Let's say I have three areas: AT, AY, and AN for TRUTH, YES, and NO, respectively. Like on the slide I will work out that Pr[YES|YES] = AT + AY and Pr[YES|NO] = AY. The ratio is thus (AT+AY)/AY = 1.1, and AT = (0.1)AY. That's the same as the slide so far.
Now I can plug in any value I want (within reason, obviously) for AY, and get the corresponding value for AT and AN. Or I can go the other way. Suppose I devote 0.25 of the spinner to NO. Then I end up with approximately 0.682 for YES and 0.0682 for NO.
Hope this helps!