Imagine that you and I are sitting at a table. Hidden in my lap, I have a jar of beans. We are going to play the traditional game wherein you try to guess the number of beans in the jar. However, you don’t get to see the jar. The rule is that I remove beans from the jar and place them on the table for as long as I like, and then at an arbitrary point ask you how many beans there are in total. That’s all you get to see.
One by one, I remove a dozen beans. As I place the twelfth bean on the table in front of you, I ask: “So how many beans are there total, including those left in the jar?”
“I have no idea,” you reasonably reply.
“Alright, well let’s try to narrow it down,” I say helpfully. “What is the greatest amount of beans I could possibly have in total?”
You reason thusly: “Well, given the Copernican principle, this twelfth bean is equally likely to fall anywhere along the distribution of the total number of beans. Thus, for example, all else held equal, there is a 50% chance that it will be within the last 50% of beans removed from the jar – or the first 50%, for that matter.
“But, obviously, it further follows that there is a 70% chance that it will be in the final 70% of beans, and a 95% chance that it will be within the last 95% of beans you might remove, and so on. In this scenario – if the 11 previous beans represent only 5% of the total – then there should be at most 11x20 total beans, or 220. Thus, I can be 95% confident that there are no more than 220 beans. Of course, the actual possible number asymptotically approaches infinity by this reasoning (say I wanted to be 99% confident?), but 95% confidence is good enough for me! So I’ll take 220 as my upper bound…”
You are wrong. I have over 1,000 beans left in the jar.
Or: you are (technically) right. There is only one bean left in the jar.
Or: any other possibility.
Either way, it seems obvious that your reasoning is completely disconnected from the actual number of beans left in the jar. Given the evidence you’ve actually seen, it seems intuitively that it could just as well be any number (12 or greater).
Where did you go wrong?
The proper Bayesian response to evidence is to pick a particular hypothesis – say, “there are fewer than 220 beans,” which is the hypothesis you just pegged at 95% confidence – and then see whether the given evidence (“he stopped at the 12th bean”) updates you towards or away from it.[1]
It seems clear that this kind of update is not what you have done in reasoning about the beans. Rather, you picked a hypothesis that was merely compatible with the evidence – “there are fewer than 220 beans.” You then found this weird value of the percentage of possible worlds wherein the evidence could possibly appear[2], out of possible worlds where the hypothesis is true (i.e. worlds where there are at least 12 beans out of worlds with fewer than 220). And this was then conflated this with the actual posterior probability of the hypothesis.
It seems to me that the Doomsday Argument is exactly analogous to this situation, except that it’s a sentient 12th bean itself (i.e. a human somewhere in the timeline) that happens to be making the guess.
I am not at all confident that I haven’t failed to address some obvious feature of the original argument. Please rebut.
[1] I’ve just tried to do this, but I’m rubbish at math, especially when it includes tricky (to me) things like ranges and summations. (Doesn’t the result depend entirely on your prior probability that there are (0, 220] beans, which would depend on your implicit upper bound for beans to begin with, if you assume there can’t be infinite beans in my jar?)
[2] Not does appear – remember, I could have stopped on any bean. This chunk of possible worlds includes, e.g. the world where I went all the way to bean 219.
His reasoning would be entirely correct if you had determined the number of beans you draw randomly from between 0 and the total number. His priors were all wrong, and so he failed.
Could we take all possible prior distributions, assign to each some prior that is probably wrong, and then use those prior distributions as theories to use the number of beans as evidence for?
(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 ;)