The paths followed by probability are not the paths of causal influence, but the paths of logical implication, which run in both directions.
Yep, that was pretty dumb. Thanks for being gentle with me.
However, I still don't understand what's wrong with my conclusion in your version of Sleeping Beauty. Upon waking, Sleeping Beauty (whichever copy of her) doesn't observe anything (colored stones or otherwise) correlated with the result of the coin flip. So it seems she has to stick with her original probability of tails having been flipped, 1/2.
Next, out of curiosity, if you had participated in my red/green thought experiment in real life, how would you anticipate if you woke up in a red room (not how would you bet, because I think IRL you'd probably care about copies of you)? I just can't even physically bring myself to imagine seeing 9,999 copies of me coming out of their respective rooms and telling me they saw red, too, when I had been so confident beforehand that this very situation would not happen. Are you anticipating in the same way as me here?
Finally, let's pull out the anthropic version of your stones in a bag experiment. Let's say someone flips an unbiased coin; if it comes up heads, you are knocked out and wake up in a white room, while if it comes up tails, you are knocked out, then copied, and one of you wakes up in a white room and the other wakes up in a black room. Let's just say the person in each room (or in just the white room if that's the only one involved) is asked to guess whether the coin came up heads or tails. Let's also say, for whatever reason, the person has resolved to, if ey wakes up in the white room, guess heads. If ey wakes up in the black room, ey won't be guessing, ey'll just be right. Now, if we repeat this experiment multiple times, with different people, it will turn out that, looking at all of the different people (/copies) that actually did wake up in white rooms, it turns out that exactly half of them will have guessed right. Right now I'm just talking about watching this experiment many times from the outside. In fact, it doesn't matter with what probability the person resolves to guess heads if ey wakes up in the white room - this result holds (that around half of the guesses from white rooms will be correct, in the long run).
Now, given all of that, here's how I would reason, from the inside of this experiment, if we're doing log scores in utils (if for some reason I didn't care about copies of me, which IRL I would) for a probability of heads. Please tell me if you'd reason differently, and why:
In a black room, duh. So let's say I wake up in a white room. I'd say, well, I only want to maximize my utility. The only way I can be sure to uniquely specify myself, now that I might have been copied, is to say that I am "notsonewuser-in-a-white-room". Saying "notsonewuser" might not cut it anymore. Historically, when I've watched this experiment, "person-in-a-white-room" guesses the coin flip correctly half of the time, no matter what strategy ey has used. So I don't think I can do better than to say 1/2. So I say 1/2 and get -1 util (as opposed to an expected -1.08496... utils which I've seen historically hold up when I look at all the people in white rooms who have said a 2/3 probability of heads).
Now I also need to explain why I think this differs from the obvious situation you brought up (obvious in that the answer was obvious, not in that it wasn't a good point to make, I think it definitely was!). For one thing, looking historically at people who pick out white stones, they have been in heads-world 2/3 of the time. I don't seem to have any other coherent answer for the difference, though, to be honest (and I've already spent hours thinking about this stuff today, and I'm tired). So my reduction's not quite done, but given the points I've made here, I don't think yours is, either. Maybe you can see flaws in my reasoning, though. Please let me know if you do.
EDIT: I think I figured out the difference. In the situation where you are simply reaching into a bag, the event "I pull out a white stone." is well defined. In the situation in which you are cloned, the event "I wake up in a white room." is only well-defined when it is interpreted as "Someone who subjectively experiences being me wakes up in a white room.", and waking up in a black room is not evidence against the truth of this statement, whereas pulling out a black stone is pretty much absolute evidence that you did not pull out a white stone.
(Crossposted from my blog)
I've been developing an approach to anthropic questions that I find less confusing than others, which I call Anthropic Atheism (AA). The name is a snarky reference to the ontologically basic status of observers (souls) in other anthropic theories. I'll have to explain myself.
We'll start with what I call the “Sherlock Holmes Axiom” (SHA), which will form the epistemic background for my approach:
Which I reinterpret as “Reason by eliminating those possibilities inconsistent with your observations. Period.” I use this as a basis of epistemology. Basically, think of all possible world-histories, assign probability to each of them according to whatever principles (eg occams razor), eliminate inconsistencies, and renormalize your probabilities. I won’t go into the details, but it turns out that probability theory (eg Bayes theorem) falls out of this just fine when you translate
P(E|H)as “portion of possible worlds consistent with H that predict E”. So it’s not really any different, but using SHA as our basis, I find certain confusing questions less confusing, and certain unholy temptations less tempting.With that out of the way, let’s have a look at some confusing questions. First up is the Doomsday Argument. From La Wik:
The article goes on to claim that “There is a 95% chance of extinction within 9120 years.” Hard to refute, but nevertheless it makes one rather uncomfortable that the mere fact of one’s existence should have predictive consequences.
In response, Nick Bostrom formulated the “Self Indication Assumption”, which states that “All other things equal, an observer should reason as if they are randomly selected from the set of all possible observers.” Applied to the doomsday argument, it says that you are just as likely to exist in 2014 in a world where humanity grows up to create a glorious everlasting civilization, as one where we wipe ourselves out in the next hundred years, so you can’t update on that mere fact of your existence. This is comforting, as it defuses the doomsday argument.
By contrast, the Doomsday argument is the consequence of the “Self Sampling Assumption”, which states that “All other things equal, an observer should reason as if they are randomly selected from the set of all actually existent observers (past, present and future) in their reference class.”
Unfortunately for SIA, it implies that “Given the fact that you exist, you should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist.” Surely that should not follow, but clearly it does. So we can formulate another anthropic problem:
This one is called the “presumptuous philosopher”. Clearly the presumptuous philosopher should not get a Nobel prize.
These questions have caused much psychological distress, and been beaten to death in certain corners of the internet, but as far as I know, few people have satisfactory answers. Wei Dai’s UDT might be satisfactory for this, and might be equivalent to my answer, when the dust settles.
So what’s my objection to these schemes, and what’s my scheme?
My objection is aesthetic; I don’t like that SIA and SSA seem to place some kind of ontological specialness on “observers”. This reminds me way too much of souls, which are nonsense. The whole “reference-class” thing rubs me the wrong way as well. Reference classes are useful tools for statistical approximation, not fundamental features of epistemology. So I'm hesitant to accept these theories.
Instead, I take the position that you can never conclude anything from your own existence except that you exist. That is, I eliminate all hypotheses that don’t predict my existence, and leave it at that, in accordance with SHA. No update happens in the Doomsday Argument; both glorious futures and impending doom are consistent with my existence, their relative probability comes from other reasoning. And the presumptuous philosopher is an idiot because both theories are consistent with us existing, so again we get no relative update.
By reasoning purely from consistency of possible worlds with observations, SHA gives us a reasonably principled way to just punt on these questions. Let’s see how it does on another anthropic question, the Sleeping Beauty Problem:
SHA says that the coin came up heads in half of the worlds, and no further update happens based on existence. I'm slightly uncomfortable with this, because SHA is cheerfully biting a bullet that has confused many philosophers. However, I see no reason not to bite this bullet; it doesn’t seem to have any particularly controversial implications for actual decision making. If she is paid for each correct guess, for example, she'll say that she thinks the coin came up tails (this way she gets $2 half the time instead of $1 half the time for heads). If she’s paid only on Monday, she’s indifferent between the options, as she should be.
What if we modify the problem slightly, and ask sleeping beauty for her credence that it’s Monday? That is, her credence that “it” “is” Monday. If the coin came up heads, there is only Monday, but if it came up tails, there is a Monday observer and a Tuesday observer. AA/SHA reasons purely from the perspective of possible worlds, and says that Monday is consistent with observations, as is Tuesday, and refuses to speculate further on which “observer” among possible observers she “is”. Again, given an actual decision problem with an actual payoff structure, AA/SHA will quickly reach the correct decision, even while refusing to assign probabilities “between observers”.
I'd like to say that we've casually thrown out probability theory when it became inconvenient, but we haven’t; we've just refused to answer a meaningless question. The meaninglessness of indexical uncertainty becomes apparent when you stop believing in the specialness of observers. It’s like asking “What’s the probability that the Sun rather than the Earth?”. That the Sun what? The Sun and the Earth both exist, for example, but maybe you meant something else. Want to know which one this here comet is going to hit? Sure I'll answer that, but these generic “which one” questions are meaningless.
Not that I'm familiar with UDT, but this really is starting to remind me of UDT. Perhaps it even is part of UDT. In any case, Anthropic Atheism seems to easily give intuitive answers to anthropic questions. Maybe it breaks down on some edge case, though. If so, I'd like to see it. In the mean time, I don’t believe in observers.
ADDENDUM: As Wei Dai, DanielLC, and Tyrrell_McAllister point out below, it turns out this doesn't actually work. The objection is that by refusing to include the indexical hypothesis, we end up favoring universes with more variety of experiences (because they have a high chance of containing *our* experiences) and sacrificing the ability to predict much of anything. Oops. It was fun while it lasted ;)