For example, if one were to ask the average person to express their credences regarding the outcome of a two horse race that they know absolutely nothing about, they will simply say "I don't know who will win" and refrain from assigning any odds, equal or otherwise. They will also tend to accept bets in which they have knowledge that the physical probabilities are 50/50 over bets that they are totally ignorant about. — sime
trying to move away from the idea that one's credence in the state H is entirely determined by the specification of the ways in which one can come to be in that state — Pierre-Normand
The thirder's position is indeed a ratio of possible words, but there is scant evidence to support the idea that credences are accurately represented by taking ratios over possible worlds. — sime
you know there will never be any new data, so that harmless prior metastasizes. — Srap Tasmaner
Even the word "state" feels too coarse for Sleeping Beauty, since it could denote the situation a robust well-defined subject finds themselves in, or it could denote the very identity of that subject. --- At least, that's how the two main camps look to me. One wonders, where am I? how did I get here? One wonders, what am I? what has made me into this? — Srap Tasmaner
Two Envelopes seems to encourage abuse of the principle of indifference in exactly this way. Maybe it's just something like this: rationality requires treating "It's one or the other but I've literally no idea which" as an uninformative 50:50 prior only when there's the real possibility of acquiring new information upon which to update that prior. I'd rather just say, no, don't do that, "I don't know" doesn't mean "It's 50:50", [...] — Srap Tasmaner
The halver's position is based on the common-sense postulate that the physical properties and propensities of objects remain unchanged when their states are conditioned upon irrelevant information. Given acceptance of that postulate plus the premise of a fair coin, why shouldn't the halver insist that they are talking about reality, as opposed to their mere mental state?
The thirder's position is indeed a ratio of possible words, but there is scant evidence to support the idea that credences are accurately represented by taking ratios over possible worlds. — sime
If a PoI advocate assigns the credence
P(Day = mon, Flip = tails ) = P(Day = mon , Flip = heads) = P(Day = tue, Flip = tails) = 1/3
Then his unconditioned credence that it is Monday is
P(Day = mon) = 2/3
which is clearly not the expression of indifference. — sime
Then for sake of consistency he must ignore the actual information he is given about the coin, by using
P(Flip = tails) = 2/3
There's no inconsistency here. It's precisely because the premise holds that the coin is equally likely to land heads or tails when tossed, and the fact that tails would result in twice as many awakenings, that Sleeping Beauty deduces that she could accept 2 to 1 odds on each awakening and break even in the long run. — Pierre-Normand
What is wrong with this conclusion of inconsistency? — sime
Some researchers are going to put you to sleep. During the [time] that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you to back to sleep with a drug that makes you forget that waking. [While you are awake], to what degree ought you believe that the outcome of the coin toss is Heads? — Adam Elga
The issue with the SB problem, is whether to consider the two potential wakings as the same experiment, or different ones. This version resolves that. SB knows that when the researchers looked at the coins, there are four possible arrangements with probability 1/4 each: {HH, HT, TH, TT}. She also knows that, since she is awake, HH is eliminated. She can update her beliefs in the other three to 1/3 each. — JeffJo
Let P(Flip | Today ) represent the conditional probability of the state of coin, relative to the value of SB's indexical "Today". Then
2/3 = P(Flip = Tails | Today = Mon Or Tue) = P(Flip = Tails) = 1/2
What is wrong with this conclusion of inconsistency? — sime
So thanks for the enlightening discussion. — Michael
Even the word "state" feels too coarse for Sleeping Beauty, since it could denote the situation a robust well-defined subject finds themselves in, or it could denote the very identity of that subject. --- At least, that's how the two main camps look to me. One wonders, where am I? how did I get here? One wonders, what am I? what has made me into this?
As you say, it's all about individuation. Lacking a fixed point of individuation, you can push the lever however you like but you won't actually move anything. — Srap Tasmaner
Thanks to you! I'll likely revisit some of your most recent objection just for the sake of clarifying my position (and refine it if needs be). Of course, I'd be delighted if you'd chime in again whenever you feel like it. — Pierre-Normand
Given these four possible experimental runs following the four possible initial coin flip results, we find that when Sleeping Beauty awakens, she can certainly rule out HH as the current state of the two coins during that specific awakening episode. However, this does not eliminate the possibility of being in either of the last two experimental runs (in addition to, of course, either of the first two). — Pierre-Normand
But honestly, all this talk of successes is irrelevant anyway. As I said before, these are two different things:
1. Sleeping Beauty's credence that the coin tossed on Sunday for the current, one-off, experiment landed heads
2. Sleeping Beauty's most profitable strategy for guessing if being asked to guess on heads or tails over multiple games
It's simply a non sequitur to argue that if "a guess of 'tails' wins 2/3 times" is the answer to the second then "1/3" is the answer to the first. — Michael
If she has amnesia she should guess heads and will will 2/3 of the time.
If she doesn't have amnesia she should guess either on the first wake up (1/2 probability so doesn't matter which she guesses) and she should guess tail with absolute certainty if she remembers having woken up before (ie on her second wake up). Again she will win 2/3 of the time.
So having amnesia or not does not change the probability that she will win, but the tactics she should use are different. — PhilosophyRunner
According to thirders, if she has amnesia then
P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday) = 1/3 — sime
So, according to thirders her probability of losing $1 when awakened on monday in a state of amnesia is 1/3.
1/3 would be her prior upon awakening and before being informed that the day is Monday. Upon being informed of this, Sleeping Beauty would update her credence to 1/2. — Pierre-Normand
Suppose that SB gets paid $1 if the coin lands tails, otherwise she must pay $1. Furthermore, suppose that before the experiment begins she is given the choice as to whether or not she will have amnesia during the course of the experiment. According to thirder reasoning, she should choose to have amnesia in order to raise the probability of tails to 2/3 — sime
The latter is obvious but not what is asked about when asked her credence that the coin landed tails. — Michael
when the researchers looked at the coins, there are four possible arrangements with probability 1/4 each: {HH, HT, TH, TT}. — JeffJo
Is/does she paid/pay this $1 on both days, or on Wednesday after the experiment is over? In the latter case, can she choose not to have amnesia, and then choose "Heads" if she recalls no other waling but change that to Tails if she does? — JeffJo
Say I draw a card. After I look at it, I tell Andy that it is a black card, Betty that it is a spade, Cindy that its value is less than 10, and David that it is a seven (all separately). I ask each what they think the probability is that it is the Seven of Spades. Andy says 1/26, Betty says 1/13, Cindy says 1/32, and David says 1/4. All are right, but that does not affect my draw. I had a 1/52 chance to draw it. — JeffJo
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