I think this is a restatement of the crux. OF COURSE the model chosen depends on the purpose of the model. For probabilities, the choice of reference class for a given prediction/measurement is key. For Sleeping Beauty specifically, the choice of whether an experientially-irrelevant wakening (which is immediately erased and has no impact) is distinct from another is a modeling choice.
Either choice for probability modeling can answer either wagering question, simply by applying the weights to the payoffs if it’s not already part of the probability
Sure. By tweaking your “weights” or other fudge factors, you can get the right answer using any probability you please. But you’re not using a generally-applicable method, that actually tells you what the right answer is. So it’s a pointless exercise that sheds no light on how to correctly use probability in real problems.
To see that the probability of Heads is not “either 1⁄2 or 1⁄3, depending on what reference class you choose, or how you happen to feel about the problem today”, but is instead definitely, no doubt about it, 1⁄3, consider the following possibility:
Upon wakening, Beauty see that there is a plate of fresh muffins beside her bed. She recognizes them as coming from a nearby cafe. She knows that they are quite delicious. She also knows that, unfortunately, the person who makes them on Mondays puts in an ingredient that she is allergic to, which causes a bad tummy ache. Muffins made on Tuesday taste the same, but don’t cause a tummy ache. She needs to decide whether to eat a muffin, weighing the pleasure of their taste against the possibility of a subsequent tummy ache.
If Beauty thinks the probability of Heads is 1⁄2, she presumably thinks the probability that it is Monday is (1/2)+(1/2)*(1/2)=3/4, whereas if she thinks the probability of Heads is 1⁄3, she will think the probability that it is Monday is (1/3)+(1/2)*(2/3)=2/3. Since 3⁄4 is not equal to 2⁄3, she may come to a different decision about whether to eat a muffin if she thinks the probability of Heads is 1⁄2 than if she thinks it is 1⁄3 (depending on how she weighs the pleasure versus the pain). Her decision should not depend on some arbitrary “reference class”, or on what bets she happens to be deciding whether to make at the same time. She needs a real probability. And on various grounds, that probability is 1⁄3.
Sure. By tweaking your “weights” or other fudge factors, you can get the right answer using any probability you please. But you’re not using a generally-applicable method, that actually tells you what the right answer is. So it’s a pointless exercise that sheds no light on how to correctly use probability in real problems.
Completely agree. The general applicable method is:
Understand what probability experiment is going on, based on the description of the problem.
Construct the sample space from mutually exclusive outcomes of this experiment
Construct the event space based on the sample space, such that it was minimal and sufficient to capture all the events that the participant of the experiment can observe
Define probability as a measure function over the event space, such that:
The sum of probabilities of events consisting of only individual mutually exclusive and collectively exshaustive outcomes was equal to 1 and
if an event has probability 1/a then this event happens on average N/a times on a repetition of probability experiment N times for any large N.
Naturally, this produce answer 1⁄2 for the Sleeping Beauty problem.
If Beauty thinks the probability of Heads is 1⁄2, she presumably thinks the probability that it is Monday is (1/2)+(1/2)*(1/2)=3/4
This is a description of Lewisian Halfism reasoning, that in incorrect for the Sleeping Beauty problem
I describe the way the Beauty is actually supposed to reason about betting scheme on a particular day here.
She needs a real probability.
Indeed. And real probability domain of function is event space, consisting of properly defined events for the probability experiment. “Today is Monday” is ill-defined in the Sleeping Beauty setting. Therefore it can’t have probability.
[ bowing out after this—I’ll read responses and perhaps update on them, but probably won’t respond (until next time) ]
To see that the probability of Heads is not “either 1⁄2 or 1⁄3, depending on what reference class you choose
I disagree. Very specifically, it’s 1⁄2 if your reference class is “fair coin flips” and 1⁄3 if your reference class is “temporary, to-be-erased experience of victims with adversarial memory problems”.
If your reference class is “wakenings who are predicting what day it is”, as the muffin variety, then 1⁄3 is a bit easier to work with (though you’d need to specify payoffs to explain why she’d EVER eat the muffin, and then 1⁄2 becomes pretty easy too). This is roughly equivalent to the non-memory-wiping wager: I’ll flip a fair coin, you predict heads or tails. If it’s heads, the wager will be $1, if it’s tails, the wager is $2. The probability of tails is not 2⁄3, but you’d pay up to $0.50 to play, right?
OK, I’ll end by just summarizing that my position is that we have probability theory, and we have decision theory, and together they let us decide what to do. They work together. So for the wager you describe above, I get probability 1⁄2 for Heads (since it’s a fair coin), and because of that, I decide to pay anything less than $0.50 to play. If I thought that the probability of heads was 0.4, I would not pay anything over $0.20 to play. You make the right decision if you correctly assign probabilities and then correctly apply decision theory. You might also make the right decision if you do both of these things incorrectly (your mistakes might cancel out), but that’s not a reliable method. And you might also make the right decision by just intuiting what it is. That’s fine if you happen to have good intuition, but since we often don’t, we have probability theory and decision theory to help us out.
One of the big ways probability and decision theory help is by separating the estimation of probabilities from their use to make decisions. We can use the same probabilities for many decisions, and indeed we can think about probabilities before we have any decision to make that they will be useful for. But if you entirely decouple probability from decision-making, then there is no longer any basis for saying that one probability is right and another is wrong—the exercise becomes pointless. The meaningful justification for a probability assignment is that it gives the right answer to all decision problems when decision theory is correctly applied.
As your example illustrates, correct application of decision theory does not always lead to you betting at odds that are naively obtained from probabilities. For the Sleeping Beauty problem, correctly applying decision theory leads to the right decisions in all betting scenarios when Beauty thinks the probability of Heads is 1⁄3, but not when she thinks it is 1⁄2.
[ Note that, as I explain in my top-level answer in this post, Beauty is an actual person. Actual people do not have identical experiences on different days, regardless of whether their memory has been erased. I suspect that the contrary assumption is lurking in the background of your thinking that somehow a “reference class” is of relevance. ]
I think this is a restatement of the crux. OF COURSE the model chosen depends on the purpose of the model. For probabilities, the choice of reference class for a given prediction/measurement is key. For Sleeping Beauty specifically, the choice of whether an experientially-irrelevant wakening (which is immediately erased and has no impact) is distinct from another is a modeling choice.
Either choice for probability modeling can answer either wagering question, simply by applying the weights to the payoffs if it’s not already part of the probability
Sure. By tweaking your “weights” or other fudge factors, you can get the right answer using any probability you please. But you’re not using a generally-applicable method, that actually tells you what the right answer is. So it’s a pointless exercise that sheds no light on how to correctly use probability in real problems.
To see that the probability of Heads is not “either 1⁄2 or 1⁄3, depending on what reference class you choose, or how you happen to feel about the problem today”, but is instead definitely, no doubt about it, 1⁄3, consider the following possibility:
If Beauty thinks the probability of Heads is 1⁄2, she presumably thinks the probability that it is Monday is (1/2)+(1/2)*(1/2)=3/4, whereas if she thinks the probability of Heads is 1⁄3, she will think the probability that it is Monday is (1/3)+(1/2)*(2/3)=2/3. Since 3⁄4 is not equal to 2⁄3, she may come to a different decision about whether to eat a muffin if she thinks the probability of Heads is 1⁄2 than if she thinks it is 1⁄3 (depending on how she weighs the pleasure versus the pain). Her decision should not depend on some arbitrary “reference class”, or on what bets she happens to be deciding whether to make at the same time. She needs a real probability. And on various grounds, that probability is 1⁄3.
Completely agree. The general applicable method is:
Understand what probability experiment is going on, based on the description of the problem.
Construct the sample space from mutually exclusive outcomes of this experiment
Construct the event space based on the sample space, such that it was minimal and sufficient to capture all the events that the participant of the experiment can observe
Define probability as a measure function over the event space, such that:
The sum of probabilities of events consisting of only individual mutually exclusive and collectively exshaustive outcomes was equal to 1 and
if an event has probability 1/a then this event happens on average N/a times on a repetition of probability experiment N times for any large N.
Naturally, this produce answer 1⁄2 for the Sleeping Beauty problem.
This is a description of Lewisian Halfism reasoning, that in incorrect for the Sleeping Beauty problem
I describe the way the Beauty is actually supposed to reason about betting scheme on a particular day here.
Indeed. And real probability domain of function is event space, consisting of properly defined events for the probability experiment. “Today is Monday” is ill-defined in the Sleeping Beauty setting. Therefore it can’t have probability.
[ bowing out after this—I’ll read responses and perhaps update on them, but probably won’t respond (until next time) ]
I disagree. Very specifically, it’s 1⁄2 if your reference class is “fair coin flips” and 1⁄3 if your reference class is “temporary, to-be-erased experience of victims with adversarial memory problems”.
If your reference class is “wakenings who are predicting what day it is”, as the muffin variety, then 1⁄3 is a bit easier to work with (though you’d need to specify payoffs to explain why she’d EVER eat the muffin, and then 1⁄2 becomes pretty easy too). This is roughly equivalent to the non-memory-wiping wager: I’ll flip a fair coin, you predict heads or tails. If it’s heads, the wager will be $1, if it’s tails, the wager is $2. The probability of tails is not 2⁄3, but you’d pay up to $0.50 to play, right?
OK, I’ll end by just summarizing that my position is that we have probability theory, and we have decision theory, and together they let us decide what to do. They work together. So for the wager you describe above, I get probability 1⁄2 for Heads (since it’s a fair coin), and because of that, I decide to pay anything less than $0.50 to play. If I thought that the probability of heads was 0.4, I would not pay anything over $0.20 to play. You make the right decision if you correctly assign probabilities and then correctly apply decision theory. You might also make the right decision if you do both of these things incorrectly (your mistakes might cancel out), but that’s not a reliable method. And you might also make the right decision by just intuiting what it is. That’s fine if you happen to have good intuition, but since we often don’t, we have probability theory and decision theory to help us out.
One of the big ways probability and decision theory help is by separating the estimation of probabilities from their use to make decisions. We can use the same probabilities for many decisions, and indeed we can think about probabilities before we have any decision to make that they will be useful for. But if you entirely decouple probability from decision-making, then there is no longer any basis for saying that one probability is right and another is wrong—the exercise becomes pointless. The meaningful justification for a probability assignment is that it gives the right answer to all decision problems when decision theory is correctly applied.
As your example illustrates, correct application of decision theory does not always lead to you betting at odds that are naively obtained from probabilities. For the Sleeping Beauty problem, correctly applying decision theory leads to the right decisions in all betting scenarios when Beauty thinks the probability of Heads is 1⁄3, but not when she thinks it is 1⁄2.
[ Note that, as I explain in my top-level answer in this post, Beauty is an actual person. Actual people do not have identical experiences on different days, regardless of whether their memory has been erased. I suspect that the contrary assumption is lurking in the background of your thinking that somehow a “reference class” is of relevance. ]