A better “outside perspective” causal diagram for the real world would be something like “coin flip on Sunday causes memory of coin flip on Monday, which causes waking or not waking on Monday and memory of the coin flip on Tuesday, which causes waking or not waking on Tuesday.” If the memories are perfectly correlated with the coin flip, they can be collapsed into one node with no losses. But now our diagram is just the coin flip causing waking or not on monday, and waking or not on tuesday.
Switching to a perspective in the middle of the experiment then requires the different days to be mutually exclusive and exhaustive, and conditions on waking rather than not waking.
This gives P(Heads|Wake)=1/3, but doesn’t have the day as a causal node.
A better “outside perspective” causal diagram for the real world would be something like “coin flip on Sunday causes memory of coin flip on Monday, which causes waking or not waking on Monday and memory of the coin flip on Tuesday, which causes waking or not waking on Tuesday.” If the memories are perfectly correlated with the coin flip, they can be collapsed into one node with no losses. But now our diagram is just the coin flip causing waking or not on monday, and waking or not on tuesday.
Switching to a perspective in the middle of the experiment then requires the different days to be mutually exclusive and exhaustive, and conditions on waking rather than not waking.
This gives P(Heads|Wake)=1/3, but doesn’t have the day as a causal node.