On every iteration we have exactly one outcome from a sample space that is realized. And every event from event space which has this outcome is also assumed to be realized. When I say “experiment” I mean a particular iteration of it yes, because one run of sleeping beauty experiment correspond to one iteration of the probability experiment. I hope it cleared the possible misunderstanding.
THE OBSERVATION IS NOT AN EVENT
Event is not an outcome, it’s a set of one or more outcomes, from the sample space, which itself has to belong to the event space.
What you mean by “observation” is a bit of a mystery. Try tabooing it—after all probability space consists of only sample space, event space and probability function, no need to invoke this extra category for no reason.
A common way to avoid rebuttal
It’s also a common way to avoid unnecessary tangents. Don’t worry we will be back to it as soon as we deal with the more interesting issue, though I suspect then you will be able to resolve your confusion yourself.
No, thats how you try to misinterpret my version to fit your incorrect model. You use the term for Elga’s one-coin version as well. Strawman arguments are another avoidance technique.
I don’t think that correcting your misunderstanding about my position can be called “strawmanning”. If anything it is unintentional strawmannig from your side, but don’t worry, no offence taken.
Yes, One-coin-version has the exact same issue, where sequential awakenings Tails&Monday, Tails Tuesday are often treated as disconnected mutually exclusive outcomes.
But anyway, it’s kind of pointless to talk about it at this point when you’ve already agreed to the the fact that the correct sample space for two coins version is {HT_HH, TT_TH, TH_TT, HH_HT}. We agree on the model, let’s see where it leads.
Huh? What does “connect these pairs” mean to pairs that I already connected?
It means that you’ve finally done the right thing of course! You’ve stopped talking about individual awakenings as if they are themselves mutually exclusive outcomes and realized that you should be talking about the pairs of sequential awakenings treating them as a single outcome of an experiment. Well done!
No, I am not.
But apparently you still don’t exactly undertand the full consequences of it. But that’s okay, you’ve already done the most difficult step, I think the rest will be easier.
I am saying that I was able to construct a valid, and useful, sample space
And indeed you did! Once again—good job! But let’s take a minute and understand what it means.
Suppose that in a particular instance of the experiment outcome TT_TH happened. What does it mean for the Beauty? It means that she is awakened the first time before the second coin was turned and then awakened the second time after the coin was turned. This outcome encompases both her awakenings.
Likewise, when outcome HT_HH happens, the Beauty is awakened before the coin turn and is not awakened after the coin turn. This outcome describes both her awakening astate and her sleeping state.
And so on with other two outcomes. Are we on the same page here?
If there was no amnesia the Beauty could easily distinguish between the outcomes where she awakes twice orr only once. But with amnesia she is none the wiser. In the moment of awakening they feel exactly the same for her.
The thing you need to properly acknowledge, is that in the probability space you’ve constructed P(Heads) doesn’t attempt to describe probability of first coin being Heads in this awakening. Once again—awakenings are not treated as outcomes themselves anymore. Now it describes probability that the coin is Heads in this iteration of experiment as a whole.
I understand, that this may be counterintuitive for you if you got accustomed to the heresy of centred possible words. This is fine—take your time. Play with the model a bit, see what kind of events you can express with it, how it relates to betting, make yourself accustomed to it. There is no rush.
I am describing how she knows that she is in either the first observation or the second.
You’ve described two pairs of mutually exclusive events.
{HT_HH, TT_TH, TH_TT}; {HH_HT} - Beauty is awakened before the coin turn; Beauty is not awakened before the coin turn
{HH_HT, TT_TH, TH_TT}; {HT_HH} - Beauty is awakened after the coin turn; Beauty is not awakened after the coin turn.
Feel free to validate that it’s indeed what these events are.
And you correctly notice that
P(Heads|HT_HH, TT_TH, TH_TT) = 1⁄3
and
P(Heads|HH_HT, TT_TH, TH_TT) = 1⁄3
Once again, I completely agree with you! This is a correct result that we can validate through a betting scheme. A Beauty that bets on Tails exclusively when she is awoken before the coin is turned is correct 66% of iterations ofexperiment. A Beauty that bets on Tails exclusively when she is awoken after the coin is turned is also correct in 66% of iterations ofexperiment. Once again, you do not have to trust me here, you are free to check this result yourself via a simulation.
And from this you assumed that the Beauty can always reason that the awakening that she is experiencing either happened before the second coin was turned or after the second coin was turned and therefore P(Heads|(HT_HH, TT_TH, TH_TT), (HH_HT, TT_TH, TH_TT)) = 1⁄3.
But this is clearly wrong, which is very easy to see.
Which is 1⁄2, because it is probability of Heads conditional on the whole sample space, where exactly 1⁄2 of the outcomes are such that the first coin is Heads. But also we may appeal to a betting argument, a Beauty that simply bets on Tails every time is correct only in 50% of experiments. This is a well known result—that per experiment betting in Sleeping Beauty should be done at 1:1 odds. But you are, nevertheless, also free to validate it yourself if you wish.
With me so far?
Now you have an opportunity to find the mistake in your reasoning yourself. It’s an actually interesting result, with fascinating consequences, by the way. And I don’t think that many people properly understand it, based on the current level of discourse about Sleeping Beauty and anthropics as a whole. So, even though it’s going to be a bit embarrasing for you, you will also discover rare and curious new piece of knowledge as a compensation for it.
On every iteration we have exactly one outcome from a sample space that is realized. And every event from event space which has this outcome is also assumed to be realized. When I say “experiment” I mean a particular iteration of it yes, because one run of sleeping beauty experiment correspond to one iteration of the probability experiment. I hope it cleared the possible misunderstanding.
Event is not an outcome, it’s a set of one or more outcomes, from the sample space, which itself has to belong to the event space.
What you mean by “observation” is a bit of a mystery. Try tabooing it—after all probability space consists of only sample space, event space and probability function, no need to invoke this extra category for no reason.
It’s also a common way to avoid unnecessary tangents. Don’t worry we will be back to it as soon as we deal with the more interesting issue, though I suspect then you will be able to resolve your confusion yourself.
I don’t think that correcting your misunderstanding about my position can be called “strawmanning”. If anything it is unintentional strawmannig from your side, but don’t worry, no offence taken.
Yes, One-coin-version has the exact same issue, where sequential awakenings Tails&Monday, Tails Tuesday are often treated as disconnected mutually exclusive outcomes.
But anyway, it’s kind of pointless to talk about it at this point when you’ve already agreed to the the fact that the correct sample space for two coins version is {HT_HH, TT_TH, TH_TT, HH_HT}. We agree on the model, let’s see where it leads.
It means that you’ve finally done the right thing of course! You’ve stopped talking about individual awakenings as if they are themselves mutually exclusive outcomes and realized that you should be talking about the pairs of sequential awakenings treating them as a single outcome of an experiment. Well done!
But apparently you still don’t exactly undertand the full consequences of it. But that’s okay, you’ve already done the most difficult step, I think the rest will be easier.
And indeed you did! Once again—good job! But let’s take a minute and understand what it means.
Suppose that in a particular instance of the experiment outcome TT_TH happened. What does it mean for the Beauty? It means that she is awakened the first time before the second coin was turned and then awakened the second time after the coin was turned. This outcome encompases both her awakenings.
Likewise, when outcome HT_HH happens, the Beauty is awakened before the coin turn and is not awakened after the coin turn. This outcome describes both her awakening astate and her sleeping state.
And so on with other two outcomes. Are we on the same page here?
If there was no amnesia the Beauty could easily distinguish between the outcomes where she awakes twice orr only once. But with amnesia she is none the wiser. In the moment of awakening they feel exactly the same for her.
The thing you need to properly acknowledge, is that in the probability space you’ve constructed P(Heads) doesn’t attempt to describe probability of first coin being Heads in this awakening. Once again—awakenings are not treated as outcomes themselves anymore. Now it describes probability that the coin is Heads in this iteration of experiment as a whole.
I understand, that this may be counterintuitive for you if you got accustomed to the heresy of centred possible words. This is fine—take your time. Play with the model a bit, see what kind of events you can express with it, how it relates to betting, make yourself accustomed to it. There is no rush.
You’ve described two pairs of mutually exclusive events.
{HT_HH, TT_TH, TH_TT}; {HH_HT} - Beauty is awakened before the coin turn; Beauty is not awakened before the coin turn
{HH_HT, TT_TH, TH_TT}; {HT_HH} - Beauty is awakened after the coin turn; Beauty is not awakened after the coin turn.
Feel free to validate that it’s indeed what these events are.
And you correctly notice that
P(Heads|HT_HH, TT_TH, TH_TT) = 1⁄3
and
P(Heads|HH_HT, TT_TH, TH_TT) = 1⁄3
Once again, I completely agree with you! This is a correct result that we can validate through a betting scheme. A Beauty that bets on Tails exclusively when she is awoken before the coin is turned is correct 66% of iterations of experiment. A Beauty that bets on Tails exclusively when she is awoken after the coin is turned is also correct in 66% of iterations of experiment. Once again, you do not have to trust me here, you are free to check this result yourself via a simulation.
And from this you assumed that the Beauty can always reason that the awakening that she is experiencing either happened before the second coin was turned or after the second coin was turned and therefore P(Heads|(HT_HH, TT_TH, TH_TT), (HH_HT, TT_TH, TH_TT)) = 1⁄3.
But this is clearly wrong, which is very easy to see.
First of all
P(Heads|(HT_HH, TT_TH, TH_TT), (HH_HT, TT_TH, TH_TT)) = P(Heads|HT_HH, TT_TH, TH_TT, HH_HT)
Which is 1⁄2, because it is probability of Heads conditional on the whole sample space, where exactly 1⁄2 of the outcomes are such that the first coin is Heads. But also we may appeal to a betting argument, a Beauty that simply bets on Tails every time is correct only in 50% of experiments. This is a well known result—that per experiment betting in Sleeping Beauty should be done at 1:1 odds. But you are, nevertheless, also free to validate it yourself if you wish.
With me so far?
Now you have an opportunity to find the mistake in your reasoning yourself. It’s an actually interesting result, with fascinating consequences, by the way. And I don’t think that many people properly understand it, based on the current level of discourse about Sleeping Beauty and anthropics as a whole. So, even though it’s going to be a bit embarrasing for you, you will also discover rare and curious new piece of knowledge as a compensation for it.