It boils down to the following question. Should she risk a single draw from
P(Flip = Heads | Today = Monday) = 1/2 (no amnesia)
Or should she risk a single draw from
P(Flip = Heads | Today = Monday Or Tuesday) = x (amnesia)
According to halfers, x = 1/2, implying that she gains nothing by choosing amnesia.
According to thirders, x = 1/3, indicating that she gains from choosing amnesia. — sime
Your suggestion that a thirder expects to gain from choosing amnesia would depend on her conflating the probability of making a correct prediction upon awakening with the frequency of the actual payout from the initial bet. — Pierre-Normand
Your suggestion that a thirder expects to gain from choosing amnesia would depend on her conflating the probability of making a correct prediction upon awakening with the frequency of the actual payout from the initial bet. — Pierre-Normand
They do indeed conflate them. Why would SB assign a credence of 1/3 to heads, if she knows
1) That the physical probability of winning the lottery is 1/2
2) That amnesia doesn't influence lottery outcomes.
3) That had she not had amnesia, then she would either know the result of the lottery on waking up, or else refer to it's probabilities as being 1/2? — sime
Since thirders make no principled distinction between causes and effects, they do indeed encourage avoidance-coping. For by their logic, one should raise one's credences for one's preferred outcomes by avoiding information.
Because her credence wasn't meant to reflect the proportion of winning bets but rather the proportion of awakening episodes that occur during an experimental run that results in the (initial) bet being won. Those two aren't the same. If Sleeping Beauty had instead been afforded the opportunity to bet on each awakening episode, then her winning frequencies would exactly match her credence that she is winning that bet. — Pierre-Normand
I think a distinction needs to be made between the probability of making a correct prediction and the frequency of making a correct prediction. That a correct prediction of tails is twice as frequent isn't that a correct prediction of tails is twice as probable – at least according to Bayesian probability.
Maybe thirders who use betting examples are simply frequentists?
Perhaps there also needs to be a distinction made between the probability of making a correct prediction and the probability of the coin having landed tails. It could be that the answers are different. This might be especially true for frequentists, as the frequency of correct predictions is not the same as the frequency of coins landing tails (there can be two correct predictions for every one coin that lands tails). — Michael
How do you suppose that we
(1) relate the random variables, whose outcomes refer to events in the B series of time, to the indexical "Today" of the A series, whose relationship to B series and to the physics of coin-tosses isn't specified by the problem? — sime
(2) How should we apply induction to "invert" the causal direction from "Today" to the state of the coin, assuming we have a satisfactory answer to (1) ?
So, from Sue's perspective (based on the exact same evidence she shares with the participant), she concludes that the coin landed tails with a 2/3 probability, despite the coin having a 1/2 propensity to land tails. Sue's credence that the coin landed tails is a consequence of both the initial propensity of the coin to land tails and the propensities of the experimental setup to place her in a room that corresponds to a tails outcome. — Pierre-Normand
How does Sue deduce that "here" indicates she is in a room occupied by a participant scheduled for two awakenings with a probability of 2/3? She draws this conclusion from the fact that twice as many such rooms exist in the facility compared to rooms allocated to participants with only one scheduled awakening. So, from Sue's perspective (based on the exact same evidence she shares with the participant), she concludes that the coin landed tails with a 2/3 probability, despite the coin having a 1/2 propensity to land tails. Sue's credence that the coin landed tails is a consequence of both the initial propensity of the coin to land tails and the propensities of the experimental setup to place her in a room that corresponds to a tails outcome. — Pierre-Normand
Two "tail" rooms refer to the same coin outcome. So when computing the probability that the coin of her room landed heads, she needs to take into consideration the other room she might have occupied. — sime
So assuming that her supervisors assign her room using a draw from a uniform distribution, her credence that the coin of her room landed heads, should be 1/2
Now suppose she doesn't make any assumptions regarding her room allocation. What then?
For all she knows, she might have been hand-picked to babysit a "heads" room. In which case the probability that her room is of a heads coin, is 1. Due to possibilities such as this, she should
1) Refrain from committing to a probability that her room is of a heads coin, until further information is given.
but nevertheless
2) Deduce that the probability of her room coin landing heads is 1/2, given that it is known information. — sime
The question, then, is whether or not the participant should reason as if they are randomly selected from the set of all participants, and so their credence that they have been or will be woken twice is 1/2. I will continue to say that they should, given the propensities of the experimental setup to place them in the position to be woken twice. — Michael
There is a twist that comes from the fact that a biconditional holds between the two propositions "E1 is now occurring" and "E2 is now occurring". How can they therefore have different probabilities of occurrence? This puzzle is solved by attending to the practical implications of establishing effective procedures for verifying their truths, or to the means of exploiting what such truths afford. — Pierre-Normand
That a correct prediction of tails is twice as frequent isn't that a correct prediction of tails is twice as probable – at least according to Bayesian probability. — Michael
Yes, an individual tails interview event is twice as probable. A tails interview where Monday ans Tuesday interviews are grouped together is equally likely as a heads interview. it comes back to the language of the question and interpretation. — PhilosophyRunner
I don't see the connection between credence in an outcome and practical implications. Proving that the optimum betting strategy over multiple games is to bet on tails doesn't verify that P(Tails|Awake) = 2/3 is true. — Michael
Following Pradeep Mutalik's argument, according to the Bayesian "Dutch Book argument", "a degree of certainty" or "degree of belief" or "credence" is essentially your willingness to wager. Specifically, if you have a "degree of certainty" of 1/n, then you should be willing to accept a bet that offers you n or more dollars for every dollar you bet.
In that case, it's not merely the expected value of the bet that determines the credence. Rather, it's your degree of certainty, 1/n, in the outcome being wagered on that makes you rationally justified in accepting a bet with such odds. — Pierre-Normand
Will you bet that the coin landed heads 100 times in a row? I wouldn't. My credence is that it almost certainly didn't land heads 100 times in a row, and that this is almost certainly my first and only interview. — Michael
Fair enough, but then a person betting that it did land on heads 100 times in a row will have a greater expected value for their winning (as long as the winnings for heads are greater than 2^100 than for tails). And their position would be the rational one. — PhilosophyRunner
Will you bet that the coin landed heads 100 times in a row? I wouldn't. My credence is that it almost certainly didn't land heads 100 times in a row, and that this is almost certainly my first and only interview. — Michael
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