By bayes rule, Pr (H | M) * Pr(X2 |H, M) / Pr(X2 |M) = Pr(H∣X2, M), which is not the same quantity you claimed to compute Pr(H∣X2).
That’s a typo. I meant to write , not .
Second, the dismissal of betting arguments is strange.
I’ll have more to say soon about what I think is the correct betting argument. Until then, see my comment in reply to Radford Neal about disagreement on how to apply betting arguments to this problem.
“probability theory is logically prior to decision theory.” Yes, this is the common view because probability theory was developed first and is easier but it’s not actually obvious this *has* to be the case.
I said logically prior, not chronologically prior. You cannot have decision theory without probability theory—the former is necessarily based on the latter. In contrast, probability theory requires no reference to decision theory for its justification and development. Have you read any of the literature on how probability theory is either an or the uniquely determined extension of propositional logic to handle degrees of certainty? If not, see my references. Neither Cox’s Theorem nor my theorem rely on any form of decision theory.
Third, dismissal of “not H and it’s Tuesday” as not propositions doesn’t make sense. Classical logic encodes arbitrary statements within AND and OR -type constructions. There isn’t a whole lot of restrictions on them.
I’ll repeat my response to Jeff Jo: The standard textbook definition of a proposition is a sentence that has a truth value of either true or false. The problem with a statement whose truth varies with time is that it does not have a simple true/false truth value; instead, its truth value is a function from time to the set {true,false}. In logical terms, such a statement is a predicate, not a proposition. For example, “Today is Monday” corresponds to the predicate . It doesn’t become a proposition until you substitute in a specific value for , e.g. “Unix timestamp 1527556491 is a Monday.”
The paradox still stands for the moment when you wake up
You have not considered the possibility that the usual decision analysis applied to this problem is wrong. There is, in fact, disagreement as to what the correct decision analysis is. I will be writing more on this in a future post.
You seem to simply declare [Beauty’s probability at the moment of awakening] to be ½, by saying:
The prior for H is even odds: Pr(H∣M)=Pr(¬H∣M)=1/2.
This is generally indistinguishable from the ½ position you dismiss that argues for that prior on the basis of “no new information.”
In fact, I explicitly said that at the instant of awakening, Beauty’s probability is the same as the prior, because at that point she does not yet have any new information. As she receives sensory input, her probability for Heads decreases asymptotically to 1⁄2. All of this is just standard probability theory, conditioning on the new information as it arrives. I dismissed the naive halfer position because it incorrectly assumes that Beauty’s sensory input is irrelevant to the determination of her probability for Heads.
You still don’t know how to handle the situation of being told that it’s Monday and needing to update your probability accordingly,
Uh, yes I do—it’s just standard probability theory again. Just do the math. If Beauty finds out that it is Monday, her new information since Sunday changes from to just , and since the problem definition assumes that
we get equal posterior probabilities for and , which is generally accepted to be the right answer.
Yes. It does not have a simple true/false truth value. Since it is sometimes true and sometimes false, its truth value is a function from time to {true, false}. That makes it a predicate, not a proposition.
It is not a fixed moment in time; if it were, the SB problem would be trivial and nobody would write papers about it. The questions about day of week and outcome of coin toss are potentially asked both on Monday and on Tuesday. This makes the rest of your analysis invalid. You keep on asserting that “today is Monday” is evaluated at a fixed moment in time, when in reality it is evaluated at at least two separate moments in time with different answers.
The sentence “the sensor detects white” is not a valid proposition; it is a predicate, because it is a function of time. Let’s write P(t) for this predicate. But yes, the sentence “the sensor detects white when you first press the button” is a legitimate proposition, precisely because specifies a particular time t for which P(t) is true, and so the truth value of the statement itself does not vary with time.
This gets us to the whole point of defining R(y,d): saying “Beauty has a stream of experiences y on day d” is as close as we can get to identifying a specific moment in time corresponding to the “this” in “this is day d”. The more nearly that y uniquely identifies the day, the more nearly that R(y,d) can be interpreted to mean “this is day d”.