Oh—I learned how, by the way. You start with some prior over how you expect the actual coins to be distributed, and then you convolute in the noise distribution of each box to get the combined distribution for each box. Then, given where the number on the outside of each box falls on the combined distribution, you can assign how much of that you expect to be signal and how much you expect to be noise by distributing improbability equally between signal and noise. Then you subtract out the expected noise.
Oh—I learned how, by the way. You start with some prior over how you expect the actual coins to be distributed, and then you convolute in the noise distribution of each box to get the combined distribution for each box. Then, given where the number on the outside of each box falls on the combined distribution, you can assign how much of that you expect to be signal and how much you expect to be noise by distributing improbability equally between signal and noise. Then you subtract out the expected noise.