Let it be not two different days but two different half-hour intervals. Or even two milliseconds—this doesn’t change the core of the issue that sequential events are not mutually exclusive.
OUTCOME: A measurable result of a random experiment.
SAMPLE SPACE: a set of exhaustive, mutually exclusive outcomes of a random experiment.
EVENT: Any subset of the sample space of a random experiment.
INDEPENDENT EVENTS: If A and B are events from the same sample space, and the occurrence of event A does not affect the chances of the occurrence of event B, then A and B are independent events.
The outside world certainly can name the outcomes {HH1_HT2, HT1_HH2, TH1_TT2, TT1_TH2}. But the subject has knowledge of only one pass. So to her, only the current pass exists, because she has no knowledge of the other pass. What happens in that interval can play no part in her belief. The sample space is {HH, HT, TH, TT}.
To her, these four outcomes represent fully independent events, because she has no knowledge of the other pass. To her, the fact that she is awake means the event {HH} has been ruled out. It is still a part of the sample space, but is is one she knows is not happening. That’s how conditional probability works; the sample space is divided into two subsets; one is consistent with the observation, and one is not.
What you are doing, is treating HH (or, in Elga’s implementation, H&Tuesday) as if it ceases to exist as a valid outcome of the experiment. So HH1_HT2 has to be treated differently than TT1_TH2, since HH1_HT2 only “exists” in one pass, while TT1_TH2 “exists” in both. This is not true. Both exist in both passes, but one is unobserved in one pass.
And this really is the fallacy in any halfer argument. They treat the information in the observation as if it applies to both days. Since H&Tuesday “doesn’t exist”, H&Monday fully represents the Heads outcome. So to be consistent, T&Monday has to fully represent the Tails outcome. As does T&Tuesday, so they are fully equivalent.
If you are observing state TH it necessary means that either you’ve already observed or will observe state TT.
You are projecting the result you want onto the process. Say I roll a six-sided die tell you that the result is odd. Then I administer the amnesia drug, and tell you that I previously told you whether th result was even or odd. I then ask you for your degree of belief that the result is a six. Should you say 1⁄6, because as far as you know the sample space is {1,2,3,4,5,6}? Or should you say 0, because “you are [now] observing a state that you’ve already observed is only {1,3,5}?
And if you try to claim that this is different because you don’t know what I told you, that is exactly the point of the Two Coin Version.
The definition of a sample space [was broken] - it’s supposed to be constructed from mutually exclusive elementary outcomes.
It is so constructed.
I’ve specifically explained how. We write down outcomes when the researcher sees the Beauty awake—when they updated on the fact of Beauty’s awakening.
Beauty is doing the updating, not “they.” She is in an experiment where there are four possible combinations for what the coins are currently showing. She has no ability to infer/anticipate what the coins were/will be showing on another day.
Her observation is that one combination, of what is in the sample space for today, is eliminated.
No, I’m not complicating this with two lists for each day. There is only one list, which documents all the awakenings of the subject,...
Maybe you missed the part where I said you can look at one, or the other, or bot as long as you don’t carry information across.
You are mistaken about what the amnesia acomplishes, Once again I send you to reread the Effects of Amnesia section.
Then you are mistaken in that section.
Her belief can be based only on what she knows. If you create a difference between the two passes, in her knowledge, then maybe you could claim a dependence. I don’t think you can in this case, but to do it requires that difference.
The Two Coin Version does not have a difference. Nothing about what she observed about the outcomes HH1_HT2 or HT1_HH2 in another pass can affect her confidence concerning them in the current pass. (And please, recall that these describe the combinations that are showing.)
The link I use to get here only loads the comments, so I didn’t find the “Effects of Amnesia” section until just now. Editing it:
“But in my two-coin case, the subject is well aware about the setting of the experiment. She knows that her awakening was based on the current state of the coins. It is derived from, but not necessarily the same as, the result of flipping them. She only knows that this wakening was based on their current state, not a state that either precedes or follows from another. And her memory loss prevents her from making any connection between the two. As a good Bayesian, she has to use only the relevant available information that can be applied to the current state.”