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Oscar_Cunningham comments on The Second Law of Thermodynamics, and Engines of Cognition - Less Wrong
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So in the following transformation:
X1Y1 -> X2Y1 X1Y2 -> X4Y1 X1Y3 -> X6Y1 X1Y4 -> X8Y1
You say that while true entropy has not increased (it stays at 2 bits), apparent entropy has, due to the observer not keeping track of X and just lumping its possible states into X2-X8. If this is the case, why doesn't observed entropy decrease as well, since phase space is preserved with the following?
X2Y1 -> X1Y1 X4Y1 -> X1Y2 X6Y1 -> X1Y3 X8Y1 -> X1Y4
(I guess DaveInNYC won't read this but I guess someone else might.)
If you lump together X's starting state into X2-X8 then you can't be sure that it isn't actually X3, X5 or X7. So you have to look at where those possibilities go as well. Then the entropy can't go down (since by Liouville's Theorem they have to go somewhere different from X2, X4, X6 and X8).