how is anthropic reasoning of this sort not generalizing from an example of 1?
That’s exactly what it is.
That says nothing about the prior probability of being born into this era, just that I was.
Wrong. A sample of 1 (the fact that you are in this era), can give you a surprisingly large amount of information about the sample space (people born, sorted by era). The chance of your sample of 1 being in the first 10^-50 of the sample space is pretty small. See also: the German tank problem
Wrong. A sample of 1 (the fact that you are in this era), can give you a surprisingly large amount of information about the sample space (people born, sorted by era). The chance of your sample of 1 being in the first 10^-50 of the sample space is pretty small. See also: the German tank problem
Does not apply. The Allies witnessed more than one tank, from which they were able to infer a sequential numbering scheme, and thus derive all sorts of information about German manufacturing capability and total armaments.
What if the Allies’ statisticians just had one representative tank sample, with the number “12” stamped on it. What could they infer then? Nothing. Maybe it’s a sequential serial number and they should infer that Germany only has ~24 tanks. Or maybe they should take as a prior that Hitler would send his older tanks into battle first, in which case you’d expect an early serial number and who knows how many tanks there are. Maybe it’s a production number and there are 12 different models of this tank class and unknown production runs of each. Maybe “12″ just means it was made in December, or from factory #12.
Ok, some of these examples are specific to the tank problem and don’t generalize to anthropic reasoning, but the point still stands: it is bad epistemology to generalize from one example. The only definitively valid anthropic conclusions is that there is at least one representative sample, not zero (i.e., that we live in a universe and at a specific time where human beings do exist). For the doomsday hypothesis, that tells us nothing.
(Another form of the anthropic principle applies to the Fermi paradox, but in this case we can infer other sample points from not seeing evidence of extraterrestrial intelligence in our local neighborhood. This is a very different line of reasoning, and does not apply here as far as I can tell.)
EDIT: To be clear, the specific point where the German tank problem analogy falls apart, is in its underlying assumption: that the tank (us) was selected at random from a pool of armaments (possible birthdays) with a uniform distribution. This is a completely unwarranted assumption. If, on the other hand, reincarnation were real and timeless, then you could look at “past” (future?) lives and start doing statistical analysis. But we don’t have that luxury, alas.
One data point cannot tell you literally nothing. If it did, then by induction, any finite number of data points would also tell you literally nothing. In most cases, a single data point tells you very little, because it is somewhat rare for people to be considering different hypotheses in which a single observed data point is orders of magnitude more likely in one hypothesis than in the other. The doomsday argument is an exception: It is 10^10 times as likely for a randomly selected human to be one of the first tenth of humans who have ever lived than it is for a randomly selected human to be one of the first 10^-11 of humans who have ever lived.
that the tank (us) was selected at random from a pool of armaments (possible birthdays) with a uniform distribution. This is a completely unwarranted assumption.
How is it warranted to assume that you are a priori much more likely to be in extremely unusual circumstances than a randomly selected human is?
An inference requires more than one data point. Let me give you a number: 7. Care to tell me what pattern this came from?
Your extra data points are summarized by your prior: you assume that your existence was randomly selected from the range of all possible human existences over all time, and then using that prior to reason about the doomsday paradox. I am saying that this prior has absolutely no rational basis (unless you are a theist and believe in [re-]incarnation).
You keep repeating this, but your only defense of it consists of examples of situations in which a single data point does not give you much information. This does not show that a single data point can never give you a significant amount of information. I have already explained how in the doomsday argument, a single data point does give you a lot of information, but your response was simply to repeat your claim and provide another example in which a single data point gives you less information.
Let me give you a number: 7. Care to tell me what pattern this came from?
Sure. 7 is a very common number for people to pick when they try to come up with an arbitrary number, so this is significant (though not overwhelming) evidence that you made up an arbitrary number, as opposed to, for instance, using a random number generator, or counting something in particular, etc. There’s been plenty of research into human random number generation, so this tells me quite a bit about what other numbers tend to be generated by the same process that you used to generate the number 7. For instance, 3 is also quite common, and even numbers and multiples of 5 are rare.
Your extra data points are summarized by your prior
I’m not sure what you mean by that. A prior shouldn’t count as a data point, and if you’re counting a prior as a data point anyway, then even in your examples of 1 data point being insufficient to draw confident conclusions from, we actually have more than 1 data point, since I have a prior about those, too.
you assume that your existence was randomly selected from the range of all possible human existences over all time, and then using that prior to reason about the doomsday paradox. I am saying that this prior has absolutely no rational basis
What do you suggest instead? Given only the information that you are a human, do you give more than 50% credence that you are one of the first 50% of humans to be born? Wouldn’t that seem rather absurd?
(unless you are a theist and believe in [re-]incarnation)
I don’t see what God or reincarnation could possibly have to do with this topic.
While I disagreed with your other comment, I’m inclined to agree with this one. “The amount of people to ever live” feels like a rather arbitrary and ill-defined number, and the argument’s Wikipedia article has a number of objections which I agree with and which follow similar lines: we don’t know that the number of people to ever live is necessarily finite, human population seems more likely to be exponentially than uniformly distributed, and the whole argument gets rather paradoxal if you apply it to itself.
It feels like the kind of thing that’s a solid argument if you accept its premises, but then there’s no particular to reason to expect that the premises would be correct.
Some philosophers have been bold enough to suggest that only people who have contemplated the Doomsday argument (DA) belong in the reference class ‘human’. If that is the appropriate reference class, Carter defied his own prediction when he first described the argument (to the Royal Society). A member present could have argued thus:
Presently, only one person in the world understands the Doomsday argument, so by its own logic there is a 95% chance that it is a minor problem which will only ever interest twenty people, and I should ignore it.
If the DA’s lifetime is governed by the principle of indifference and the Copernican principle then based on the length of its current existence, and assuming that it is randomly drawn from a reference class of probabilistic speculations it is 95% certain that it will be refuted before the year 2500.
If the DA is not itself subject to these principles then its assumption that the human race’s survival-time can be modeled using them appears to be a paradox (to Lansberg & Dewynne).
Hm, interesting. It is a bit unfair to talk about what the first person to formulate the doomsday argument could have used it to predict, for the same reason it is unfair to talk about what the first people to exist could have used the doomsday argument to predict: being the person who comes up with an idea that becomes popular is rare, and we expect probabilistic arguments to fail in rare edge cases. Also, the self-referencing doomsday argument rebuttal offers a counter-rebuttal of itself: It is more recent and has been considered by fewer people than the doomsday argument has, so it will probably have a shorter lifespan. (yes, I thought of the obvious counter-counter-rebuttal.)
The main point that I get out of those examples is that the Doomsday Argument is really a fully general argument that can be applied to pretty much anything. You can apply it to itself, or I could apply it to predict how many days of life I still have left, or for how long I will continue to remain employed (either at my current job, or in general), or to how many LW comments I am yet to write...
A claim like “my daughter just had her first day of school, and if we assume that she’s equally likely to find herself in any position n of her total amount of lifetime days in school N, then it follows that there’s a 95% chance that she will spend a maximum of 20 days of her life going to school” would come off as obviously absurd, but I’m not sure why the Doomsday Argument would be essentially any different.
It’s possible to argue that it is appropriate to use SIA in some of those examples, but SSA for the duration of the human race.
“my daughter just had her first day of school, and if we assume that she’s equally likely to find herself in any position n of her total amount of lifetime days in school N, then it follows that there’s a 95% chance that she will spend a maximum of 20 days of her life going to school”
The doomsday argument doesn’t say that, even if you do use SSA with the reference class of days that your daughter is in school. You’re confusing the likelihood of the evidence given the hypothesis with the posterior probability of the hypothesis given the evidence.
It feels like the kind of thing that’s a solid argument if you accept its premises, but then there’s no particular to reason to expect that the premises would be correct.
That summarized it better than I could. Thank you!
You are assuming a uniform distribution across the sample space. That assumption seems to be hanging in mid-air: there is absolutely nothing supporting it.
Not assuming. Defining. The measure on the sample space that I was referring to when I said “first 10^-50” is the probability distribution on the sample space. No other measure of the sample space has even been mentioned. I really was not saying anything that’s not totally tautological.
Edit: Or are you referring back to the Doomsday argument and suggesting that you are not, a priori, a randomly selected human?
The measure on the sample space that I was referring to when I said “first 10^-50” is the probability distribution on the sample space.
That’s not what people usually mean when they say things like “the first 1% of the sample space”. When they want to talk about the probability distribution on the sample space they tend to say “the first 1% of the population”.
How do you know what the probability distribution in the sample space is, anyway?
I was not trying to imply that the probability distribution on the sample space was known, simply that whatever it is, the probability of being in the first 10^-50 of it is 10^-50. The entire point of the doomsday argument (and much of Bayesian statistics, for that matter) is that if the probability distribution on the sample space is not known, you can use your data point to update your priors over them.
That’s exactly what it is.
Wrong. A sample of 1 (the fact that you are in this era), can give you a surprisingly large amount of information about the sample space (people born, sorted by era). The chance of your sample of 1 being in the first 10^-50 of the sample space is pretty small. See also: the German tank problem
Does not apply. The Allies witnessed more than one tank, from which they were able to infer a sequential numbering scheme, and thus derive all sorts of information about German manufacturing capability and total armaments.
What if the Allies’ statisticians just had one representative tank sample, with the number “12” stamped on it. What could they infer then? Nothing. Maybe it’s a sequential serial number and they should infer that Germany only has ~24 tanks. Or maybe they should take as a prior that Hitler would send his older tanks into battle first, in which case you’d expect an early serial number and who knows how many tanks there are. Maybe it’s a production number and there are 12 different models of this tank class and unknown production runs of each. Maybe “12″ just means it was made in December, or from factory #12.
Ok, some of these examples are specific to the tank problem and don’t generalize to anthropic reasoning, but the point still stands: it is bad epistemology to generalize from one example. The only definitively valid anthropic conclusions is that there is at least one representative sample, not zero (i.e., that we live in a universe and at a specific time where human beings do exist). For the doomsday hypothesis, that tells us nothing.
(Another form of the anthropic principle applies to the Fermi paradox, but in this case we can infer other sample points from not seeing evidence of extraterrestrial intelligence in our local neighborhood. This is a very different line of reasoning, and does not apply here as far as I can tell.)
EDIT: To be clear, the specific point where the German tank problem analogy falls apart, is in its underlying assumption: that the tank (us) was selected at random from a pool of armaments (possible birthdays) with a uniform distribution. This is a completely unwarranted assumption. If, on the other hand, reincarnation were real and timeless, then you could look at “past” (future?) lives and start doing statistical analysis. But we don’t have that luxury, alas.
One data point cannot tell you literally nothing. If it did, then by induction, any finite number of data points would also tell you literally nothing. In most cases, a single data point tells you very little, because it is somewhat rare for people to be considering different hypotheses in which a single observed data point is orders of magnitude more likely in one hypothesis than in the other. The doomsday argument is an exception: It is 10^10 times as likely for a randomly selected human to be one of the first tenth of humans who have ever lived than it is for a randomly selected human to be one of the first 10^-11 of humans who have ever lived.
How is it warranted to assume that you are a priori much more likely to be in extremely unusual circumstances than a randomly selected human is?
What I said was:
An inference requires more than one data point. Let me give you a number: 7. Care to tell me what pattern this came from?
Your extra data points are summarized by your prior: you assume that your existence was randomly selected from the range of all possible human existences over all time, and then using that prior to reason about the doomsday paradox. I am saying that this prior has absolutely no rational basis (unless you are a theist and believe in [re-]incarnation).
You keep repeating this, but your only defense of it consists of examples of situations in which a single data point does not give you much information. This does not show that a single data point can never give you a significant amount of information. I have already explained how in the doomsday argument, a single data point does give you a lot of information, but your response was simply to repeat your claim and provide another example in which a single data point gives you less information.
Sure. 7 is a very common number for people to pick when they try to come up with an arbitrary number, so this is significant (though not overwhelming) evidence that you made up an arbitrary number, as opposed to, for instance, using a random number generator, or counting something in particular, etc. There’s been plenty of research into human random number generation, so this tells me quite a bit about what other numbers tend to be generated by the same process that you used to generate the number 7. For instance, 3 is also quite common, and even numbers and multiples of 5 are rare.
I’m not sure what you mean by that. A prior shouldn’t count as a data point, and if you’re counting a prior as a data point anyway, then even in your examples of 1 data point being insufficient to draw confident conclusions from, we actually have more than 1 data point, since I have a prior about those, too.
What do you suggest instead? Given only the information that you are a human, do you give more than 50% credence that you are one of the first 50% of humans to be born? Wouldn’t that seem rather absurd?
I don’t see what God or reincarnation could possibly have to do with this topic.
While I disagreed with your other comment, I’m inclined to agree with this one. “The amount of people to ever live” feels like a rather arbitrary and ill-defined number, and the argument’s Wikipedia article has a number of objections which I agree with and which follow similar lines: we don’t know that the number of people to ever live is necessarily finite, human population seems more likely to be exponentially than uniformly distributed, and the whole argument gets rather paradoxal if you apply it to itself.
It feels like the kind of thing that’s a solid argument if you accept its premises, but then there’s no particular to reason to expect that the premises would be correct.
I don’t understand. What does it mean to apply the doomsday argument to itself?
(from here)
(from here)
Hm, interesting. It is a bit unfair to talk about what the first person to formulate the doomsday argument could have used it to predict, for the same reason it is unfair to talk about what the first people to exist could have used the doomsday argument to predict: being the person who comes up with an idea that becomes popular is rare, and we expect probabilistic arguments to fail in rare edge cases. Also, the self-referencing doomsday argument rebuttal offers a counter-rebuttal of itself: It is more recent and has been considered by fewer people than the doomsday argument has, so it will probably have a shorter lifespan. (yes, I thought of the obvious counter-counter-rebuttal.)
The main point that I get out of those examples is that the Doomsday Argument is really a fully general argument that can be applied to pretty much anything. You can apply it to itself, or I could apply it to predict how many days of life I still have left, or for how long I will continue to remain employed (either at my current job, or in general), or to how many LW comments I am yet to write...
A claim like “my daughter just had her first day of school, and if we assume that she’s equally likely to find herself in any position n of her total amount of lifetime days in school N, then it follows that there’s a 95% chance that she will spend a maximum of 20 days of her life going to school” would come off as obviously absurd, but I’m not sure why the Doomsday Argument would be essentially any different.
It’s possible to argue that it is appropriate to use SIA in some of those examples, but SSA for the duration of the human race.
The doomsday argument doesn’t say that, even if you do use SSA with the reference class of days that your daughter is in school. You’re confusing the likelihood of the evidence given the hypothesis with the posterior probability of the hypothesis given the evidence.
That summarized it better than I could. Thank you!
I think you’re wrong. If you have no idea of what sample space looks like, that chance isn’t “pretty small”, it’s unknown.
The chance of being in the first 10^-50 of the sample space is, in fact, precisely 10^-50, which is pretty small.
You are assuming a uniform distribution across the sample space. That assumption seems to be hanging in mid-air: there is absolutely nothing supporting it.
Not assuming. Defining. The measure on the sample space that I was referring to when I said “first 10^-50” is the probability distribution on the sample space. No other measure of the sample space has even been mentioned. I really was not saying anything that’s not totally tautological.
Edit: Or are you referring back to the Doomsday argument and suggesting that you are not, a priori, a randomly selected human?
That’s not what people usually mean when they say things like “the first 1% of the sample space”. When they want to talk about the probability distribution on the sample space they tend to say “the first 1% of the population”.
How do you know what the probability distribution in the sample space is, anyway?
If no other measure on the sample space has been mentioned, they aren’t likely to be referring to anything else. Anyway, it’s what I was referring to.
You don’t, but whatever it is, you have a 10^-50 chance of being in the first 10^-50 of it.
Then I don’t understand what was the point of your original comment.
I said: “If you have no idea of what sample space looks like, that chance isn’t “pretty small”, it’s unknown.”
Your example where you happen to know not just the sample space but actually the probability distribution seems entirely irrelevant.
I was not trying to imply that the probability distribution on the sample space was known, simply that whatever it is, the probability of being in the first 10^-50 of it is 10^-50. The entire point of the doomsday argument (and much of Bayesian statistics, for that matter) is that if the probability distribution on the sample space is not known, you can use your data point to update your priors over them.