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.

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[-]Manfred11y00

So, there's a few differences between these two diagrams, but they are all very similar differences. The two main ones are whether the day is an independent causal node from the coin flip, and whether not waking up on Tuesday is an event. The first difference implies the second but not vice versa, so it's the stronger condition and the second is the weaker.

I think it's not hard to answer these questions. But I also want to develop guidelines for what to do in other problems.

Here is what I consider a slam dunk against diagram 1: Before the experiment, what is Sleeping Beauty's probability that on Tuesday, she doesn't wake up? 1/2.

If she has a probability for this thing, she must be representing it as an event. This is that second difference I mentioned. In fact, the whole reason the probability changes upon waking in diagram 2 is because this non-waking option gets eliminated and its probability mass evenly redistributed.

If diagrams 1 and 2 were the only two options, this would sort of be the end - but they're not, they're just two diagrams that we promoted to attention by some other process. We want to be able to help out the intuitive process that fits these causal models to this story problem.

The categories to keep track of are the possible events, the causal structure, and the constraints on the various nodes (at least for simple problems like this where those things don't change based on observations).

When you update on observations in these simple problems, this doesn't mean changing the causal diagram to eliminate that observation from the causal diagram. Instead, you leave your information about causation unchanged and update just by conditionalization.

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[-]Manfred11y00

Hmm, maybe one can't eliminate the [???] node. This is because non-causal-diagram information, such as observations, is conditioned on for all nodes, in the ordinary non-causal-diagram way. So the information encoded in diagram 1 really is causal, even if it's unphysical. One could interpret this as a heuristic argument against diagram 1 - no unphysical causal nodes.

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[-]Manfred11y00

In general, you can remove the [???] node. Causal information can just be about how the wake on (day) node's value is determined by the coin flip, it doesn't necessitate another node. And even though this doesn't fully determine the value of the wake on (day) node, our information still determines our probabilities.

Though if the causal picture of the universe is also true, the different undetermied choices are caused by either an extra causal factor ([???]) or un-tracked differences in the coin flip - just like how the different outcomes of the coin flip are caused by small differences in initial conditions.

If all coin flips are treated the same, this forces the decision to be made by some other sort of initial conditions. And it is a peculiarity of the Sleeping Beauty problem that this would be unphysical for diagram 1 - if we go back to the marble game, there's any number of physical processes that work.

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