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timtyler comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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If you are locked in a lead-lined box, the answer to question "is it night time outside now" varies over time - even though you learn nothing new.
Similarly with Beauty, as she moves through the experimental procedure.
But here you've put the time-indexical "now" into your description of the event. You're asking for P(it is night, now). In the Beauty case question asked is what is P(heads), now. In the first case every moment that goes by we're talking about a temporally distinct event. You're actually asking about a different event every moment- so it isn't surprising that the answer changes from moment to moment. The Sleeping Beauty problem is always about the same event.
The coin flip doesn't change - but Beauty does. She goes in one end of the expertiment and comes out the other side, and she knows roughly where she is on that timeline. Probabalities are subjective - and in this example we are asked for Beauty's "credence" - i.e. her subjective probability at a particular point in time. That's a function of the observer, not just the observed.
Yes. But subjective probability is a function of the information someone has not where they are on the time-line. Which is why people keep asking what information Beauty is updating on. We're covering 101 stuff at this point.
...and going round in circles, I might note. We did already discuss the issue of exactly when Beauty updates close by - here.
Also, we already know where we differ. We consider "subjective probability" to refer to different things. Given your notion of "subjective probability", your position makes perfect sense, IMO. I just don't think that is how scientists generally use the term.
It's hard to tell but I'm not sure your notion of "subjective probability" is coherent- specifically because you keep talking about different events depending on what time you're in. That sounds like a recipe for disaster. But alright.
Does this mean we can just agree to specify payouts in our probability problems from now on? Or must we now investigate which one of us is using the term the way scientists do? Unfortunately this disagreement suggest to me that scientists may not know exactly what they mean by subjective probability.
Subjective probability is a basic concept in decision theory. Scientists have certainly tried hard to say exactly what they mean by the term. E.g. see this one, from 1963:
"A Definition of Subjective Probability" - F. J. Anscombe; R. J. Aumann
Sure. I don't see anything in there to suggest that subjective probability isn't time symmetrical (by which I mean that a subjective probability regarding an event can be held at any time and there is not reason for the probability to change unless the person's evidence changes). Can you do a better job formalizing what your alternative is?
Except she doesn't. She'll give the same answer on Monday as she will on Tuesday, because she doesn't learn anything by waking up.