I think halfers would say from Beauty’s perspective P(Heads) and P(Heads|I am awake now) means the same thing. Any reasoning done by Beauty is based on the fact that “I am awake now”.
Yes, at first glance it seems natural to assume this, but I see rejecting that claim as the only consistent way of developing the halfer view.
Any reasoning done by Beauty is based on the fact that “I am awake now”
There’s a sense in which that’s true, which is that the fact that Beauty is awake is available for beauty to use in whatever calculations they perform. However, this isn’t the say as claiming they have to use it a particular way.
P(Heads|I am awake now) doesn’t just mean “I am awake now” is available in beauty’s databank, but also indicates that instead of sampling coin flips we’re sampling times when beauty is awake. So that’s the suble slight of hand; the invisible shift in assumptions.
In the end I do agree they are answering different questions. But still think halfers in some sense are more subjective.
The halfer position poorly developed is more subjective. Although maybe most halfers develop it that way, I don’t know.
P(Heads|I am awake now) doesn’t just mean “I am awake now” is available in beauty’s databank, but also indicates that instead of sampling coin flips we’re sampling times when beauty is awake. So that’s the suble slight of hand; the invisible shift in assumptions.
This is what I think, thus our disagreement: Why should indexicals such as now, today or in some other anthropic problems I or myself, be considered as a sample at all? In my opinion they should be regarded as fundamentals/irreducibles. Their meanings are primitively understood from a given perspective. And the perspective disagreement is caused by this irreducible nature. I tried to explain my idea in this post. And that’s why I don’t think we should throwout the “no new information” argument too early.
My idea is by no means popular. And I am not trying to persuade you in accepting it. Just want to show the possibility that some halfer argument can be regarded as more “subjective based”. Whether they are poorly developed or not, I don’t know. Maybe time will tell.
Take now as an example. I suggest it refers to the moment most immediate to the perception and subjective experience. A primitively understood concept that cannot be further reduced. So “I’m awake now” is a just a logical truth, i.e. no new information.
While traditionally people like to treat now as a randomly sampled moment during the experiment. As you did in the previous reply stating by using “I am awake now” indicates we are sampling times. People disagree about the exact sampling method (SSA or SIA). But I am suggesting now should not be regarded as a random sample at all.
We agree that the only objective chance in this setup is (a) the coin and (b) the choice of Alex’s waking day, right? So that’s a sample space of 4 outcomes, each equally probable a priori.
Alex and Beauty can communicate only when they are both awake. For both of them, this eliminates the outcome {Heads,AlexWakesOnTuesday}, leading both to the conclusion that Pr(Heads|both awake) = 13. No disagreement.
What you are describing, 4 equal probable priori in terms of Monday/Tuesday is thirder’s logic. Then Beauty doesn’t even have to check on Alex, by realizing I’m awake beauty would eliminate (Heads, Tuesday) already. Seeing Alex changes nothing. Thirderism will not lead to any perspective disagreement.
One the other hand Halfers suggest finding “I am awake today” is to be expected (no new information) due to the experiment setup of one guaranteed awakening. Combining the status of Alex they will say the 4 equal probable priori are (Heads, Alex is awake today), (Tails, Alex is awake today),(Heads, Alex is asleep today), (Tails, Alex is asleep today). Finding Alex awake eliminate the last two. The probability of Heads remains unchanged at 1⁄2. Again seeing Alex changes nothing.
If you are suggesting Halfers should change the probability of Heads to 1⁄3 after seeing Alex awake then I don’t see how that can be coherent. Just adding an independent component (Alex’s random awakening) should not give Beauty additional information about the coin toss. Similar idea has been brought by Michael Titelbaum’s Technicolor Beauty and Radford Neal’s FNC. Both suggest Thirderism (or almost Thirderism in the case of FNC) is correct instead of Halfers.
No, am I not Thirding. Thirders consider Beauty’s wake-ups on Monday and Tuesday to be “events”. That’s not what I’m doing here.
- - - -
We’ll need to be a bit more specific about your Alex & Beauty set-up: I hope you don’t mind if we specify how Alex’s waking day is chosen. Let’s say it’s chosen to be Monday if a fair die roll turns out Odd, or Tuesday if Even.
Now, you must agree that there are precisely two independent sources of objective chance in your experiment: the coin and the die roll. There are four equiprobable outcomes, of course:
Pr(Heads,Odd) = 14
Pr(Heads,Even) = 14
Pr(Tails,Odd) = 14
Pr(Tails,Even) = 14
When Beauty sees Alex awake, she must discount the outcome (Heads,Even) from the sample space, leaving Pr(H,O) = Pr(T,O) = Pr(T,E) = 13. I really hope that everyone should agree that those are the correct posterior objective chances.
- - - -
This analysis says nothing about Beauty’s subjective credence for Heads, nor does it attempt to justify any philosophical issues arising. We can have that philosophical debate later. But let’s ground the discussion in something that’s not open to interpretation by at least agreeing on the objective chances.
I think using dice roll to decide Alex’s awakening is an excellent idea.
Back to the topic at hand, what I have been arguing is that the correct halfer argument should not be based on objective chance. Thus my disagreement with Chris so far. I think it should instead be based on the irreducible nature of perspectives, that it roots in the subjective. That’s why I brought up the perspective disagreement in the first place.
What you have argued: that after seeing Alex awake Beauty should eliminate one of the 4 events, reflects the fact that Alex has met Beauty in an awakening. But it does not use the whole information Beauty has: she specifically knows Alex has met Beauty in this awakening. And the specification of this awakening is based on her perspective, as it is the one which she has subjective experience of.
You said you are not thirding, only considering the objective chance. I believe you. But I was saying by considering the objective chance this way one has to endorse Thirderism. For example, if Beauty checks on Alex and see him asleep instead, by the same logic she must discount (Heads, Odd), leaving P(H,E) = P(T,O) = P(T,E) = 1⁄3. So it doesn’t matter whether Beauty finds Alex awake or asleep, she will say P(H)=1/3 regardless. Then Beauty should’ve regard 1⁄3 as the correct answer even before looking at Alex anyway. (This argument is due to Michael Titelbaum ’s Technicolor Beauty)
So do I think the objective chance you presented are correct? Yes, I do. Just as Beauty should think Alex’s answer of 1⁄3 is correct. Do I think Beauty should give the same answer of 1/3? No, I don’t. Because at the end of the day I am a halfer, and I believe perspectives are fundamental, the disagreement between Alex and Beauty is valid.
So do I think the objective chance you presented are correct? Yes, I do.
That’s good, I just wanted to establish that baseline. I think it’s healthy for everyone to confirm their understanding of the objective chances before moving on to debating philosophical issues of credence. Thank you.
“But I was saying by considering the objective chance this way one has to endorse Thirderism”
No. If I asked you to calculate the objective chances then I hope you (and everyone else here) would perform exactly the same calculation. Presented as a maths problem, as I did, it says nothing about my (or your, or anyone else’s) philosophical leanings.
Perhaps you meant that by equating the objective Pr(...) with Credence one has to be a Thirder?
I do like the example though. It confused me enough that I felt I had to post again (replying to myself … bad form). What follows is pure speculation.
I think there’s something here about scopes/closures.
Within the scope of a single wake-up, “Alex is awake at the same time as me” is admissible as an event: specifically the subset { (H,O), (T,O), (T,E) } of the sample space.
Within the scope of the whole experiment, “Alex is awake at the same time as me” is inadmissible as an event: it is not a well-defined subset of the sample space.
So for games whose payout decomposes additively over wake-ups, you may consider each wake-up in isolation and, if you do so, you must use all available events including Alex’s sleeping state, Thirder-style.
For games whose payout does not decompose additively over wake-ups, you must consider the experiment as a whole and, in doing so, find that you have no new information, Halfer-style.
Hey, don’t worry about replying to oneself. I do it quite often :)
I just realized the root cause of our debate after reading this:
That’s good, I just wanted to establish that baseline. I think it’s healthy for everyone to confirm their understanding of the objective chances before moving on to debating philosophical issues of credence. Thank you.
You regard “objective chance” as the baseline, any philosophical leaning is built on top of that. But I think the philosophical aspect of the problem cuts deeper than that.I regard (postulates) perspective reasoning as fundamental and objective reasoning as derived. While in most people recognize (postulate) objective reasoning (i.e. perspective-independent reasoning) as fundamental and indexicals as additional information that need to be incorporated on top of objective reasoning. That’s why I call my approach Perspective Based Reasoning.
So when seeing “objective chance” I was not considering it as the baseline, but treating it as “thinking from a god’s eye view” (which give the same answer as Alex’s perspective). To me it is no more fundamental than thinking from Beauty’s first-person viewpoint. Just two different systems of reasoning. And the two perspective gives different answers, both are valid, thus the disagreement.
It is why when you ask what should Beauty think about the objective chance I keep stressing what should Beauty think from her perspective. You want to leave the philosophical leanings out of it for now, but to me using objective chance as the baseline already has a philosophical leaning.
It is also why I don’t think halfers should change the probability to 1⁄3 after seeing Alex awake. I think the best rebuttal is still the one I just discussed. By the same logic Beauty would change the probability to 1⁄3 after seeing Alex asleep as well. Since it is going to be 1⁄3 regardless of Alex’s status her probability should have been 1⁄3 right from the start, i.e. after waking up.
I don’t think we’re a million miles apart on this. We can both tolerate Halfing and Thirding, in your case because they’re both reasonable perspectives and in my case because it doesn’t matter anyway. Your characterisation of our disagreement seems about right: let’s come back to that in the comments of some future unsuspecting author’s post on Sleeping Beauty :)
Yes, at first glance it seems natural to assume this, but I see rejecting that claim as the only consistent way of developing the halfer view.
There’s a sense in which that’s true, which is that the fact that Beauty is awake is available for beauty to use in whatever calculations they perform. However, this isn’t the say as claiming they have to use it a particular way.
P(Heads|I am awake now) doesn’t just mean “I am awake now” is available in beauty’s databank, but also indicates that instead of sampling coin flips we’re sampling times when beauty is awake. So that’s the suble slight of hand; the invisible shift in assumptions.
The halfer position poorly developed is more subjective. Although maybe most halfers develop it that way, I don’t know.
This is what I think, thus our disagreement: Why should indexicals such as now, today or in some other anthropic problems I or myself, be considered as a sample at all? In my opinion they should be regarded as fundamentals/irreducibles. Their meanings are primitively understood from a given perspective. And the perspective disagreement is caused by this irreducible nature. I tried to explain my idea in this post. And that’s why I don’t think we should throwout the “no new information” argument too early.
My idea is by no means popular. And I am not trying to persuade you in accepting it. Just want to show the possibility that some halfer argument can be regarded as more “subjective based”. Whether they are poorly developed or not, I don’t know. Maybe time will tell.
I’m confused. I don’t think I understand what it means to regard them as irreducibles instead of a sample?
Take now as an example. I suggest it refers to the moment most immediate to the perception and subjective experience. A primitively understood concept that cannot be further reduced. So “I’m awake now” is a just a logical truth, i.e. no new information.
While traditionally people like to treat now as a randomly sampled moment during the experiment. As you did in the previous reply stating by using “I am awake now” indicates we are sampling times. People disagree about the exact sampling method (SSA or SIA). But I am suggesting now should not be regarded as a random sample at all.
We agree that the only objective chance in this setup is (a) the coin and (b) the choice of Alex’s waking day, right? So that’s a sample space of 4 outcomes, each equally probable a priori.
Alex and Beauty can communicate only when they are both awake. For both of them, this eliminates the outcome {Heads,AlexWakesOnTuesday}, leading both to the conclusion that Pr(Heads|both awake) = 13. No disagreement.
What you are describing, 4 equal probable priori in terms of Monday/Tuesday is thirder’s logic. Then Beauty doesn’t even have to check on Alex, by realizing I’m awake beauty would eliminate (Heads, Tuesday) already. Seeing Alex changes nothing. Thirderism will not lead to any perspective disagreement.
One the other hand Halfers suggest finding “I am awake today” is to be expected (no new information) due to the experiment setup of one guaranteed awakening. Combining the status of Alex they will say the 4 equal probable priori are (Heads, Alex is awake today), (Tails, Alex is awake today),(Heads, Alex is asleep today), (Tails, Alex is asleep today). Finding Alex awake eliminate the last two. The probability of Heads remains unchanged at 1⁄2. Again seeing Alex changes nothing.
If you are suggesting Halfers should change the probability of Heads to 1⁄3 after seeing Alex awake then I don’t see how that can be coherent. Just adding an independent component (Alex’s random awakening) should not give Beauty additional information about the coin toss. Similar idea has been brought by Michael Titelbaum’s Technicolor Beauty and Radford Neal’s FNC. Both suggest Thirderism (or almost Thirderism in the case of FNC) is correct instead of Halfers.
No, am I not Thirding. Thirders consider Beauty’s wake-ups on Monday and Tuesday to be “events”. That’s not what I’m doing here.
- - - -
We’ll need to be a bit more specific about your Alex & Beauty set-up: I hope you don’t mind if we specify how Alex’s waking day is chosen. Let’s say it’s chosen to be Monday if a fair die roll turns out Odd, or Tuesday if Even.
Now, you must agree that there are precisely two independent sources of objective chance in your experiment: the coin and the die roll. There are four equiprobable outcomes, of course:
Pr(Heads,Odd) = 14
Pr(Heads,Even) = 14
Pr(Tails,Odd) = 14
Pr(Tails,Even) = 14
When Beauty sees Alex awake, she must discount the outcome (Heads,Even) from the sample space, leaving Pr(H,O) = Pr(T,O) = Pr(T,E) = 13. I really hope that everyone should agree that those are the correct posterior objective chances.
- - - -
This analysis says nothing about Beauty’s subjective credence for Heads, nor does it attempt to justify any philosophical issues arising. We can have that philosophical debate later. But let’s ground the discussion in something that’s not open to interpretation by at least agreeing on the objective chances.
I think using dice roll to decide Alex’s awakening is an excellent idea.
Back to the topic at hand, what I have been arguing is that the correct halfer argument should not be based on objective chance. Thus my disagreement with Chris so far. I think it should instead be based on the irreducible nature of perspectives, that it roots in the subjective. That’s why I brought up the perspective disagreement in the first place.
What you have argued: that after seeing Alex awake Beauty should eliminate one of the 4 events, reflects the fact that Alex has met Beauty in an awakening. But it does not use the whole information Beauty has: she specifically knows Alex has met Beauty in this awakening. And the specification of this awakening is based on her perspective, as it is the one which she has subjective experience of.
You said you are not thirding, only considering the objective chance. I believe you. But I was saying by considering the objective chance this way one has to endorse Thirderism. For example, if Beauty checks on Alex and see him asleep instead, by the same logic she must discount (Heads, Odd), leaving P(H,E) = P(T,O) = P(T,E) = 1⁄3. So it doesn’t matter whether Beauty finds Alex awake or asleep, she will say P(H)=1/3 regardless. Then Beauty should’ve regard 1⁄3 as the correct answer even before looking at Alex anyway. (This argument is due to Michael Titelbaum ’s Technicolor Beauty)
So do I think the objective chance you presented are correct? Yes, I do. Just as Beauty should think Alex’s answer of 1⁄3 is correct. Do I think Beauty should give the same answer of 1/3? No, I don’t. Because at the end of the day I am a halfer, and I believe perspectives are fundamental, the disagreement between Alex and Beauty is valid.
That’s good, I just wanted to establish that baseline. I think it’s healthy for everyone to confirm their understanding of the objective chances before moving on to debating philosophical issues of credence. Thank you.
No. If I asked you to calculate the objective chances then I hope you (and everyone else here) would perform exactly the same calculation. Presented as a maths problem, as I did, it says nothing about my (or your, or anyone else’s) philosophical leanings.
Perhaps you meant that by equating the objective Pr(...) with Credence one has to be a Thirder?
I do like the example though. It confused me enough that I felt I had to post again (replying to myself … bad form). What follows is pure speculation.
I think there’s something here about scopes/closures.
Within the scope of a single wake-up, “Alex is awake at the same time as me” is admissible as an event: specifically the subset { (H,O), (T,O), (T,E) } of the sample space.
Within the scope of the whole experiment, “Alex is awake at the same time as me” is inadmissible as an event: it is not a well-defined subset of the sample space.
So for games whose payout decomposes additively over wake-ups, you may consider each wake-up in isolation and, if you do so, you must use all available events including Alex’s sleeping state, Thirder-style.
For games whose payout does not decompose additively over wake-ups, you must consider the experiment as a whole and, in doing so, find that you have no new information, Halfer-style.
Hey, don’t worry about replying to oneself. I do it quite often :)
I just realized the root cause of our debate after reading this:
You regard “objective chance” as the baseline, any philosophical leaning is built on top of that. But I think the philosophical aspect of the problem cuts deeper than that.I regard (postulates) perspective reasoning as fundamental and objective reasoning as derived. While in most people recognize (postulate) objective reasoning (i.e. perspective-independent reasoning) as fundamental and indexicals as additional information that need to be incorporated on top of objective reasoning. That’s why I call my approach Perspective Based Reasoning.
So when seeing “objective chance” I was not considering it as the baseline, but treating it as “thinking from a god’s eye view” (which give the same answer as Alex’s perspective). To me it is no more fundamental than thinking from Beauty’s first-person viewpoint. Just two different systems of reasoning. And the two perspective gives different answers, both are valid, thus the disagreement.
It is why when you ask what should Beauty think about the objective chance I keep stressing what should Beauty think from her perspective. You want to leave the philosophical leanings out of it for now, but to me using objective chance as the baseline already has a philosophical leaning.
It is also why I don’t think halfers should change the probability to 1⁄3 after seeing Alex awake. I think the best rebuttal is still the one I just discussed. By the same logic Beauty would change the probability to 1⁄3 after seeing Alex asleep as well. Since it is going to be 1⁄3 regardless of Alex’s status her probability should have been 1⁄3 right from the start, i.e. after waking up.
I don’t think we’re a million miles apart on this. We can both tolerate Halfing and Thirding, in your case because they’re both reasonable perspectives and in my case because it doesn’t matter anyway. Your characterisation of our disagreement seems about right: let’s come back to that in the comments of some future unsuspecting author’s post on Sleeping Beauty :)