I have had an intuition of my own for why and I agree with you about this now. The result in my previous comment is wrong.
Before, it seemed correct that and I used this when calculating and .
But now I believe that is wrong which is why I got the wrong result.
Why do I believe this?
The martingale condition/conservation of expected evidence applies only if the relevant epistemic state is exactly the same before and after observing the evidence. By “epistemic state” I mean the set of statements the agent knows is true. On Sunday SB knows she is the original, on Monday she does not. So on Sunday her epistemic state contains the statement “I am the original”, on Monday it does not. This change of epistemic state before observing evidence “awake” and after observing evidence “awake” doesn’t allow the application of the martingale as you described. Because if she learns she is NOT the original, then her probabilities for Heads and Tails differ from the priors. For the martingale to apply, additionally she has to know on Sunday that she will know she is the original on Monday. Which is not the case.
According to my calculations and .
From your second post:
Based on the above, I believe the correct math would be:
So on Sunday her epistemic state contains the statement “I am the original”, on Monday it does not.
Epistemic states can change freely without breaking martingales; on Sunday, she’s in room 1; put her to sleep and wake her up in room 1 or 2, randomly chosen; no duplications. Her epistemic state has moved from “I’m in room 1” to “I don’t know what room I’m in”, and yet the martingale works perfectly across this.
In any case, the statement is not “I am the original” but “on Monday, I was the original”, which is preserved.
I’ve also weakened the requirements; now it’s the law of total probability that is violated, not the martingale.
If you get , you’re violating simple Bayes on the initial Sunday, which forces Q to equal the prior on worlds.
I realized some of my mistakes since my previous post.
Not to get into overmuch details, I’ll just say I’m back at where I started.
I believe:
.
and .
In the context of your second post:
and .
If “simple Bayes” implies that upon learning she is the original on Tuesday SB must have the same posteriors as she had priors on Sunday when she also knew she was original, then I don’t see why should I accept “simple Bayes” as a property of probability.
If she learns on Tuesday “I am the original” is false, then her probability for Tails becomes 1, which makes this positive evidence for Tails. Therefore, learning on Tuesday “I am the original” is true must be negative evidence for Tails. If she doesn’t lower her probability for Tails upon learning she is the original, that would be violation of conservation of expected evidence. In other words, confirmation bias—no matter what she learns on Tuesday (out of available options), her probability for Tails may go up but never down.
Simple Bayes means that, if there is no longer any anthropic angle to your problem, you should be able to apply non-anthropic updating.
Equivalently-ish[1] it implies that it doesn’t matter when you started thinking anthropically; if you started doing it yesterday and then updated on one day’s observation, it would be the same as if you started doing it today, or twenty years ago and updated on twenty years of observations. And it also doesn’t matter that you could have been X, if you know that you’re not X.
Formally, this is anchor-free: there isn’t an anchor moment that defines the beginning of when you’re thinking anthropically[2].
Anchor-free is stronger than simple Bayes; simple Bayes is just the equivalent of anchor-free in situations where there is no anthropic uncertainty. Anchor-free is the real condition I want, but simple Bayes is enough for the impossibility result.
One needs to be careful here; technically, any method with a specified anchor is “anchor-free” in a certain sense if it says “whenever you actually started thinking anthropically, then go back to this moment in the past (or future) and anchor then”. If the specified anchor is part of the definition of the method, then it’s “anchor-free” in that it formally doesn’t matter when you started thinking anthropically, because in all cases you just go to a specified anchor[3] and role out everything backwards/forwards from that I don’t consider “anchor specified” to be anchor-free.
“Go to a specified anchor” is not actually that trivial to define. If the “anchor specified” truly behaves like “anchor-free” then it has to work from the very moment of first consciousness (as that’s certainly a place you could start anchoring from), when you could have been, potentially, any being in the universe. So there’s a universal anthropic probability that depends potentially on the details of all agents who ever were or ever could be.
I have had an intuition of my own for why
Before, it seemed correct that
But now I believe that
Why do I believe this?
The martingale condition/conservation of expected evidence applies only if the relevant epistemic state is exactly the same before and after observing the evidence. By “epistemic state” I mean the set of statements the agent knows is true. On Sunday SB knows she is the original, on Monday she does not. So on Sunday her epistemic state contains the statement “I am the original”, on Monday it does not. This change of epistemic state before observing evidence “awake” and after observing evidence “awake” doesn’t allow the application of the martingale as you described. Because if she learns she is NOT the original, then her probabilities for Heads and Tails differ from the priors. For the martingale to apply, additionally she has to know on Sunday that she will know she is the original on Monday. Which is not the case.
According to my calculations
From your second post:
Based on the above, I believe the correct math would be:
Epistemic states can change freely without breaking martingales; on Sunday, she’s in room 1; put her to sleep and wake her up in room 1 or 2, randomly chosen; no duplications. Her epistemic state has moved from “I’m in room 1” to “I don’t know what room I’m in”, and yet the martingale works perfectly across this.
In any case, the statement is not “I am the original” but “on Monday, I was the original”, which is preserved.
I’ve also weakened the requirements; now it’s the law of total probability that is violated, not the martingale.
If you get
I realized some of my mistakes since my previous post.
Not to get into overmuch details, I’ll just say I’m back at where I started.
I believe:
In the context of your second post:
If “simple Bayes” implies that upon learning she is the original on Tuesday SB must have the same posteriors as she had priors on Sunday when she also knew she was original, then I don’t see why should I accept “simple Bayes” as a property of probability.
If she learns on Tuesday “I am the original” is false, then her probability for Tails becomes 1, which makes this positive evidence for Tails. Therefore, learning on Tuesday “I am the original” is true must be negative evidence for Tails. If she doesn’t lower her probability for Tails upon learning she is the original, that would be violation of conservation of expected evidence. In other words, confirmation bias—no matter what she learns on Tuesday (out of available options), her probability for Tails may go up but never down.
Simple Bayes means that, if there is no longer any anthropic angle to your problem, you should be able to apply non-anthropic updating.
Equivalently-ish [1] it implies that it doesn’t matter when you started thinking anthropically; if you started doing it yesterday and then updated on one day’s observation, it would be the same as if you started doing it today, or twenty years ago and updated on twenty years of observations. And it also doesn’t matter that you could have been X, if you know that you’re not X.
Formally, this is anchor-free: there isn’t an anchor moment that defines the beginning of when you’re thinking anthropically [2] .
Anchor-free is stronger than simple Bayes; simple Bayes is just the equivalent of anchor-free in situations where there is no anthropic uncertainty. Anchor-free is the real condition I want, but simple Bayes is enough for the impossibility result.
One needs to be careful here; technically, any method with a specified anchor is “anchor-free” in a certain sense if it says “whenever you actually started thinking anthropically, then go back to this moment in the past (or future) and anchor then”. If the specified anchor is part of the definition of the method, then it’s “anchor-free” in that it formally doesn’t matter when you started thinking anthropically, because in all cases you just go to a specified anchor [3] and role out everything backwards/forwards from that I don’t consider “anchor specified” to be anchor-free.
“Go to a specified anchor” is not actually that trivial to define. If the “anchor specified” truly behaves like “anchor-free” then it has to work from the very moment of first consciousness (as that’s certainly a place you could start anchoring from), when you could have been, potentially, any being in the universe. So there’s a universal anthropic probability that depends potentially on the details of all agents who ever were or ever could be.