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vinayak comments on Open Thread: March 2010 - Less Wrong
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I have two basic questions that I am confused about. This is probably a good place to ask them.
What probability should you assign as a Bayesian to the answer of a yes/no question being yes if you have absolutely no clue about what the answer should be? For example, let's say you are suddenly sent to the planet Progsta and a Sillpruk comes and asks you whether the game of Doldun will be won by the team Strigli.
Consider the following very interesting game. You have been given a person who will respond to all your yes/no questions by assigning a probability to 'yes' and a probability to 'no'. What's the smallest sequence of questions you can ask him to decide for sure that a) he is not a rationalist, b) he is not a Bayesian?
This is somewhat similar to the question I asked in Reacting to Inadequate Data. It was hit with a -3 rating though... so apparently it wasn't too useful.
The consensus of the comments was that the correct answer is .5.
Also of note is Bead Jar Guesses and its sequel.
If you truly have no clue, .5 yes and .5 no.
Ah, but here you have some clues, which you should update on, and knowing how is much trickier. One clue is that the unkown game of Doldun could possibly have more than 2 teams competing, of which only 1 could win, and this should shift the probabilities in favor of "No". How much? Well that depends on your probability distribution for an unknown game to have n competing teams. Of course, there may be other clues that should shift the probabilty towards "yes".
But the game of Doldun could also have the possibility of cooperative wins. Or it could be unwinnable. Or Strigli might not be playing. Or Strigli might be the only team playing - it's the team against the environment! Or Doldun could be called on account of a rain of frogs. Or Strigli's left running foobar could break a chitinous armor plate and be replaced by a member of team Baz, which means that Baz gets half credit for a Strigli win.
All of which means that you shouldn't be too confident in your probability distribution in such a foreign situation, but you still have to come up with a probability if it's relevant at all for action. Bad priors can hurt, but refusal to treat your uncertainty in a Bayes-like fashion hurts more (with high probability).
Yes, but in this situation you have so little information that .5 doesn't seem remotely cautious enough. You might as well ask the members of Strigli as they land on Earth what their probability is that the Red Sox will win at a spelling bee next year - does it look obvious that they shouldn't say 50% in that case? .5 isn't the right prior - some eensy prior that any given possibly-made-up alien thing will happen, adjusted up slightly to account for the fact that they did choose this question to ask over others, seems better to me.
Unless there's some reason that they'd suspect it's more likely for us to ask them a trick question whose answer is "No" than one whose question is "Yes" (although it is probably easier to create trick questions whose answer is "No", and the Striglian could take that into account), 50% isn't a bad probability to assign if asked a completely foreign Yes-No question.
Basically, I think that this and the other problems of this nature discussed on LW are instances of the same phenomenon: when the space of possibilities (for alien culture, Omega's decision algorithm, etc) grows so large and so convoluted as to be utterly intractable for us, our posterior probabilities should be basically our ignorance priors all over again.
It seems to me that even if you know that there is a Doldun game, played by exactly two teams, of which one is Strigli, which game exactly one team will entirely win, 50% is as high as you should go. If you don't have that much precise information, then 50% is an extremely generous upper bound for how likely you should consider a Strigli win. The space of all meaningful false propositions is hugely larger than the space of all meaningful true propositions. For every proposition that is true, you can also contradict it directly, and then present a long list of indirectly contradictory statements. For example: it is true that I am sitting on a blue couch. It is false that I am not on a blue couch - and also false that I am on a red couch, false that I am trapped in carbonite, false that I am beneath the Great Barrier Reef, false that I'm in the Sea of Tranquility, false that I'm equidistant between the Sun and the star Polaris, false that... Basically, most statements you can make about my location are false, and therefore the correct answer to most yes-or-no questions you could ask about my location is "no".
Basically, your prior should be that everything is almost certainly false!
The odds of a random sentence being true are low, but the odds of the alien choosing to give you a true sentence are higher.
A random alien?
No, just a random alien that (1) I encountered and (2) asked me a question.
The two conditions above restrict enormously the general class of “possible” random aliens. Every condition that restricts possibilities brings information, though I can't see a way of properly encoding this information as a prior about the answer to said question.
[ETA:] Note that I don't necessarily accept cousin_it's assertion, I just state my interpretation of it.
Well, let's say I ask you whether all "fnynznaqre"s are "nzcuvovna"s. Prior to using rot13 on this question (and hypothesizing that we hadn't had this particular conversation beforehand), would your prior really be as low as your previous comment implies?
(Of course, it should probably still be under 50% for the reference class we're discussing, but not nearly that far under.)
Given that you chose this question to ask, and that I know you are a human, then screening off this conversation I find myself hovering at around 25% that all "fnynznaqre"s are "nzcuvovna"s. We're talking about aliens. Come on, now that it's occurred to you, wouldn't you ask an E.T. if it thinks the Red Sox have a shot at the spelling bee?
Yes, but I might as easily choose a question whose answer was "Yes" if I thought that a trick question might be too predictable of a strategy.
1/4 seems reasonable to me, given human psychology. If you expand the reference class to all alien species, though, I can't see why the likelihood of "Yes" should go down— that would generally require more information, not less, about what sort of questions the other is liable to ask.
But it is true that you are not on a red couch.
Negation is a one-to-one map between true and false propositions.
Since you can understand the alien's question except for the nouns, presumably you'd be able to tell if there was a "not" in there?
Yes, you have made a convincing argument, I think, that given that a proposition does not involve negation, as in the alien's question, that it is more likely to be false than true. (At least, if you have a prior for being presented with questions that penalize complexity. The sizes of the spaces of true and false propositions, however, are the same countable infinity.) (Sometimes I see claims in isolation, and so miss that a slightly modified claim is more correct and still supports the same larger claim.)
ETA: We should also note the absence of any disjunctions. It is also true that Alicorn is sitting on a blue couch or a red couch. (Well, maybe not, some time has passed since she reported sitting on a blue couch. But that's not the point.)
This effect may be screened off if, for example, you have a prior that the aliens first choose whether the answer should be yes or no, and then choose a question to match the answer.
That the aliens chose to translate their word as the English 'game' says, I think, a lot.
"Game" is one of the most notorious words in the language for the virtual impossibility of providing a unified definition absent counterexamples.
A family resemblance is still a resemblance.
Could you include a source for this quote, please?
Googling it would've told you that it's from Wittgenstein's Philosophical Investigations.
"A game is a voluntary attempt to overcome unnecessary obstacles."
This is, perhaps, a necessary condition but not a sufficient one. It is true of almost all hobbies, but I wouldn't classify hobbies such as computer programming or learning to play the piano as games.
I wouldn't class most hobbies as attempts to overcome unnecessary obstacles either -- certainly not playing a musical instrument, where the difficulties are all necessary ones. I might count bird-watching, of the sort where the twitcher's goal is to get as many "ticks" (sightings of different species) as possible, as falling within the definition, but for that very reason I'd regard it as being a game.
One could argue that compulsory games at school are a counterexample to the "voluntary" part. On the other hand, Láadan has a word "rashida": "a non-game, a cruel "playing" that is a game only for the dominant "player" with the power to force others to participate [ra=non- + shida=game]". In the light of that concept, perhaps these are not really games for the children forced to participate.
But whatever nits one can pick in Bernard Suits' definition, I still think it makes a pretty good counter to Wittgenstein's claims about the concept.
Hm. For actual aliens I don't think even that's justified, without either knowing more about their psychology, or having some sort of equally problematic prior regarding the psychology of aliens.
I was conditioning on the probability that the question is in fact meaningful to the aliens (more like "Will the Red Sox win the spelling bee?" than like "Does the present king of France's beard undertake differential diagnosis of the psychiatric maladies of silk orchids with the help of a burrowing hybrid car?"). If you assume they're just stringing words together, then there's not obviously a proposition you can even assign probability to.
Hey, maybe they're Zen aliens who always greet strangers by asking meaningless questions.
More sensibly, it seems to me roughly equally plausible that they might ask a meaningful question because the correct answer is negative, which would imply adjusting the prior downward; and unknown alien psychology makes me doubtful of making a sensible guess based on context.
For #2, I don't see how you could ever be completely sure the other was rationalist or Bayesian, short of getting their source code; they could always have one irrational belief hiding somewhere far from all the questions you can think up.
In practice, though, I think I could easily decide within 10 questions whether a given (honest) answerer is in the "aspiring rationalist" cluster and/or the "Bayesian" cluster, and get the vast majority of cases right. People cluster themselves pretty well on many questions.
For two, can I just have an extended preface that describes a population, an infection rate for some disease and a test with false positivity rates and false negativity rates and see if the person gives me the right answer?
1: If you have no information to support either alternative more than the other, you should assign them both equal credence. So, fifty-fifty. Note that yes-no questions are the easiest possible case, as you have exactly two options. Things get much trickier once it's not obvious what things should be classified as the alternatives that should be considered equally plausible.
Though I would say that in this situation, the most rational approach would be to tell the Sillpruk, "I'm sorry, I'm not from around here. Before I answer, does this planet have a custom of killing people who give the wrong answer to this question, or is there anything else I should be aware of before replying?"
2: This depends a lot how we define a rationalist and a Bayesian. A question like "is the Bible literally true" could reveal a lot of irrational people, but I'm not certain of the amount of questions that'd need to be asked before we could know for sure that they were irrational. (Well, since 1 and 0 aren't probabilities, the strict answer to this question is "it can't be done", but I'm assuming you mean "before we know with such a certainty that in practice we can say it's for sure".)
Yes, I should be more specific about 2.
So let's say the following are the first three questions you ask and their answers -
Q1. Do you think A is true? A. Yes. Q2. Do you think A=>B is true? A. Yes. Q3. Do you think B is true? A. No.
At this point, will you conclude that the person you are talking to is not rational? Or will you first want to ask him the following question.
Q4. Do you believe in Modus Ponens?
or in other words,
Q4. Do you think that if A and A=>B are both true then B should also be true?
If you think you should ask this question before deciding whether the person is rational or not, then why stop here? You should continue and ask him the following question as well.
Q5. Do you think that if you believe in Modus Ponens and if you also think that A and A=>B are true, then you should also believe that B is true as well?
And I can go on and on...
So the point is, if you think asking all these questions is necessary to decide whether the person is rational or not, then in effect any given person can have any arbitrary set of beliefs and he can still claim to be rational by adding a few extra beliefs to his belief system that say the n^th level of "Modus Ponens is wrong" for some suitably chosen n.
I think that belief in modus ponens is a part of the definition of "rational", at least practically. So Q1 is enough. However, there are not much tortoises among the general public, so this type of question isn't probably much helpful.