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The Maximum Entropy Distribution of a Circuit
So suppose you just have your circuit is just two wires. It takes two inputs and it just outputs both of them. Then you're like, the maximum entropy distribution is 5050 over both wires. And then suppose you just randomly added an and gate to that circuit. You still output A and B, but you've just adding an and gate as some auxiliary computation for no reason. The probability of C being on has to be like the probability of A being on times the probability of B being on plus like the interaction term. So naively it won't because your maximum entropy distribution will just be like A, B, C, all 5050 and independent.