Silent Acorns

Lombrosity is correct. The selection of the "60 billion ball" is not random.

Given your setup, the odds are 50-50. All you know is that the numbers 1-2-3-4-5-6-7 have been printed.

Bayesian analysis doesn't apply here because you know that the printer is still printing. It has to be in order for it to be analagous to the question "are we doomed?".

<edit: actually, Bayesian analysis doesn't apply because the selection of the number is not random. You defined it as the most recent number to come off the printer>

Let me put it this way. Your scenario would be appropriate for bayesian analysis it went like this :

1) the printer printed one number per page. On page 1 it prtinted a 1, on page 2, etc.
2) A coin is flipped to determine if a stack of 10 blank pages or 1 000 000 pages in inserted into the printer
2) You are brought to the room, the printer has finished printing
3) A man hands you a random page from the ouput pile.
4) The number is 7.

Under this scenario, you are justified in saying, the stack used probably had only 10 pages.

But this scenario is different from the doomsday scenario, where the printer must be still printing, by definition.

Jesse

I don't agree the printer needs to still be printing for the analogy to work. A videotape of the printer printing, with the point where you start watching the tape selected randomly by the experimenter, works better as an analogy, if you agree with the self-sampling assumption.

I'll ask you the same question I've been asking others: in the "two batches of opposite sex" scenario, if all the humans in all the batches are created at the same time (and with a coinflip determining whether there are 3 males and 5000 females or vice versa), then would you agree it makes sense to say the odds are 5000:3 that the larger batch is the same sex as me? If so, do you think it would make a difference if instead of all the humans being created at once, they were created in sequence, with the first 3 being one sex and the next 5000 being the opposite sex? In both cases it will be equally true that if everyone in this experiment bets that the larger batch is the same sex as themselves, 5000 will end up being correct and only 3 will end up being incorrect; whether the people in both batches are created all at once or in sequence seems completely irrelevant to me.

Jesse

Maybe a statistics class, although I think Bayesian reasoning is also used a lot in A.I. and computerized data analysis (see this LA Times article on 'Bayesian Networks' in A.I., or this page for some additional links on the subject). Here's a page that gives a basic outline of Bayesian inference:

http://www.statisticalengineering.co...s_thinking.htm

Silent Acorns

Perhaps it would be clearer if I put it this way:

1) Suppose an urn is filled with one ball for each person to ever live (past and future)
2) The urn contains either 100 billion, or 100 trillion balls based on a random coin toss
3) One ball is selected at random for each person
4) Each person enters the world in order based on their number
5) By adding up all the people before you, you know your number is 60 billion

How many balls were in the urn?

Bayesian analysis suggests that the answer is almost certainly 100 billion, so the question becomes,

1) Is Bayesian analysis appropriate for this scenario?

and,

2) Is this scenario descriptive of real life?

The answer to (1) is yes. The selection of the individual observation was random. However, the the answer to (2) is no. My existance at this time and place is not the result of some cosmic random selection. I am number 60 000 000 000 because I came after number 59 999 999 999. Just like page 7 comes after page 6.

The problem with the "doomsday argument" is that any attempt to make the process Bayesian, takes the scenario out of reality, and any attempt to ground the scenario in reality makes it non-Bayesian.

Jesse

Silent Acorns:
The answer to (1) is yes. The selection of the individual observation was random. However, the the answer to (2) is no. My existance at this time and place is not the result of some cosmic random selection. I am number 60 000 000 000 because I came after number 59 999 999 999. Just like page 7 comes after page 6.

I still say that's irrelevant. The self-sampling assumption just says you should reason as if you are the result of a random selection from all observers, because if everyone reasons that way, most of them will get the right answer.

You didn't answer my question about the two-batches-with-opposite-sexes experiment, and whether it would make any difference at all whether the people in the batches were created all at once or in sequence. If you wish to continue this discussion, please address it.

Silent Acorns

Yes it does. If the printer has stopped printing, and the last number printed is shown to you, you will either see 10 or 1 million. If the printer is still printing and you are shown that the last number printed is 7, all you've learned is that the printer has just started printing.

Also, the end of the print job is supposed to be analagous to the end of humanity. As a human, I have to make my observation during the humanity's existance (i.e. while the printer is printing).
Unfortunately, this scenario doesn't match with reality. My birth order in the "great human chain" was not randomly selected by an outside experimenter. My birth order is not random, and therefore, non-Bayesian.
I agree with you, but I don't see the conection to the OP.

Silent Acorns

Let's analyze this idea. Given the following scenario:

A) 50% chance that only 100 billion people will ever live
B) 50% chance that 100 trillion people will live

Suppose you must bet on how many humans will live. If you are correct you win $100, if you're wrong you lose $100. If everyone followed the "self-sampling assumption" how would things turn out?

Using the doomsday method, the first 100 billion people would bet on A (they supposedly have a 99.9% chance of being correct)and the rest would bet on B (they're sure to win - if they exist).

If A is true, everyone wins $100. The average is +$100.00.
If B is true, 999 people win $100 for every person who loses $100. The average is +$99.80.
Either way, the average person wins.

But suppose we limit our analysis to the first 100 billion only. How do they fair? After all, they are the ones the doomsday scenario refers to.

If A is true everyone wins $100.
If B is true everyone loses $100.

Overall, they have a 50% chance of winning $100 and a 50% chance of losing $100. In other words, for the first 100 billion people, it's a 50-50 bet.

Jesse

No, the doomsday argument should be used by everyone. Remember that in reality, a realistic prior probability distribution would not be a binary choice between two possible civilization-lengths, but would probably have some finite probability for any number of people, so there's no upper limit beyond which you can look at your birth rank and know with 100% certainty how many more people are going to be born. Regardless of what distribution you use--a binary one or a more realistic one--you should find that the average expected returns will be maximized if everyone uses the same type of Bayesian reasoning. In the binary-choice distribution, if the population is large than a lot of people using this type of reasoning will have 100% certainty about how many more people will be born, but that's a special case, a weird consequence of using such a binary-choice distribution in the first place.

You still haven't told me your opinion about the two-batches scenario, in both the all-people-created-at-once version and the people-created-sequentially version. Again, please do so if you want to continue the discussion.

Silent Acorns

But the question is, "what are the odds that humanity will end at the 100 billionth person?" and "does knowing that we are the 60 billionth person change the odds?"

The first question is a matter of calculating the odds of variou sprobabilities. The answer to the scond question is an emphatic "NO!". The fact is that Bayesian analysis is not appropriate here, at least not in the manner the Doomsdayers use it. My statistical meta-analysis proves it. If the doomdayers' method was correct it would give the same answer as my analysis. My analysis is correct, therefore the doomsday method is wrong.

Let me alter the scenario a little:

There are 20 worlds, all independent of each other.
A) in 1 only 100 billion people will ever live
B) in 19 100 trillion people will live

Now ofer the following bet to the 60 billionth person born to each world, select one of the following:
1) he gets $1000 if he is on the A world, $0 if not
2) he gets $100 if he is on a B world, $0 if not

If the doomsday statistics are correct, there is a 98% chance that he is on an A world. He should therefore take bet (1). However, if he ignores the fact that he's the 60 billionth person, he should take (2). How do the two stratagies play out, from the point of view of an outside observer?

Doomsdayers:
19 will be wrong, 1 will be correct: total winnings = $1000

non-Doomsdayers:
19 will be correct, 1 will be wrong: total winnings = $1900

The non-Doomsdayers win.
This is irrelevant for two reasons.
1) we're talking about this simple scenario
2) If the doomsday method doesn't work for this simple scenario, it won't work for more complicated ones
Yes I did, but I guess you missed it. I siad I agree with you, but I don't see the connection to the doomsday scenario.

If this is about the Self-sampling assumption, the answer is that the doomsday application of Bayesian analysis is wrong. The correct way to apply the self-sampling assumption is to assume you are a random selection of 20 worlds, and knowig that you are the 60 billionth person is irrelevant.

Jesse

Silent Acorns:
But the question is, "what are the odds that humanity will end at the 100 billionth person?" and "does knowing that we are the 60 billionth person change the odds?"

The first question is a matter of calculating the odds of variou sprobabilities. The answer to the scond question is an emphatic "NO!". The fact is that Bayesian analysis is not appropriate here, at least not in the manner the Doomsdayers use it. My statistical meta-analysis proves it. If the doomdayers' method was correct it would give the same answer as my analysis. My analysis is correct, therefore the doomsday method is wrong.

Let me alter the scenario a little:

There are 20 worlds, all independent of each other.
A) in 1 only 100 billion people will ever live
B) in 19 100 trillion people will live

Now ofer the following bet to the 60 billionth person born to each world, select one of the following:
1) he gets $1000 if he is on the A world, $0 if not
2) he gets $100 if he is on a B world, $0 if not

If the doomsday statistics are correct, there is a 98% chance that he is on an A world. He should therefore take bet (1). However, if he ignores the fact that he's the 60 billionth person, he should take (2). How do the two stratagies play out, from the point of view of an outside observer?

Doomsdayers:
19 will be wrong, 1 will be correct: total winnings = $1000

non-Doomsdayers:
19 will be correct, 1 will be wrong: total winnings = $1900

There's an important qualification to the Doomsday argument which you may not know about--if there are multiple worlds that all obey about the same known prior probability distribution, these other worlds cancel out the effect of one's birth rank, so one should not modify the prior probability in this case.

Suppose God makes a choice between two possible worlds, one with 100 billion and one with 100 trillion people, and picks randomly to decide which version to create with a 5% chance of the first and a 95% chance of the second. God also decides that these humans will be the only intelligent beings ever created in the entire cosmos. In that case, a person who observes that their birth rank is less than 100 billion should use Bayesian reasoning to conclude there's a 98% chance God ended up creating only 100 billion people. But if God instead decides to create a whole bunch of different worlds, 95% of which go extinct after 100 trillion people and 5% of which go extinct after 100 billion, a person who observes their birth ranks to be under 100 billion does not learn anything new, and should still stick with the prior 5% probability of being on a world with only 100 billion people. After all, for any specific birth rank under 100 billion--say, 50 billion--if you look throughout the cosmos at all the people who had that birth rank, 5% of them will be on worlds where 100 billion were born and 95% of them will be on worlds where 100 trillion were born. So even if you agree with the self-sampling assumption, in a universe where multiple worlds following the same prior probability distribution exist, the self-sampling assumption tells you you shouldn't modify that prior distribution when guessing how long intelligent life on your own world.

Of course, this only works if the prior probability distribution represents the actual probability that civilization will last various lengths. If the prior distribution just represents your best guess as to the probability distribution, one which could be wrong, then taking into account your own birth rank should still lead you to modify this prior, even if there are multiple instantiations. For example, if God was creating a bunch of 100 billion worlds and a bunch of 100 trillion worlds but I didn't know the frequency of each, if 95%-5% was just my best guess, then observing my own birth rank to be less than 100 billion should lead me to change my guess about the frequencies of the two types of worlds.

This difference between taking the prior as the actual known probability and taking it as something more subjective is an important philosophical question in one's approach to the whole notion of "probability"--people practicing Bayesian inference, especially when applied to fields like artificial intelligence, usually go the more subjective route. See the link I gave Christopher Lords earlier:

http://www.statisticalengineering.co...s_thinking.htm


Bostrom discusses the "no outsiders" requirement in his dissertation (second link on this page), especially p. 123 under "objection 2" (skip to the paragraph which begins, ‘The second reason for the doomsayer not to grant a probability shift in the above example is that the no-outsider requirement is not satisfied.’)

Jesse:
Remember that in reality, a realistic prior probability distribution would not be a binary choice between two possible civilization-lengths,

Silent Acorns:
This is irrelevant for two reasons.
1) we're talking about this simple scenario
2) If the doomsday method doesn't work for this simple scenario, it won't work for more complicated ones

But it does work in this simple scenario, provided you assume that if the population is more than 100 billion, you will include all the people with birth rank above 100 billion in your expected returns calculation, as opposed to ruling them out as you tried to do. And the reason to do this is clear when you compare the binary-choice scenario to more realistic scenarios, and see that the binary-choice scenario is just a special case where a number of observers may be 100% sure about how many more people will be born, as opposed to just 99% sure or 80% sure as some might be in slightly more realistic distributions (and surely you would include such people in the expected returns, no?)

Jesse:
You still haven't told me your opinion about the two-batches scenario, in both the all-people-created-at-once version and the people-created-sequentially version. Again, please do so if you want to continue the discussion.

Silent Acorns:
Yes I did, but I guess you missed it. I siad I agree with you, but I don't see the connection to the doomsday scenario.

If this is about the Self-sampling assumption, the answer is that the doomsday application of Bayesian analysis is wrong. The correct way to apply the self-sampling assumption is to assume you are a random selection of 20 worlds, and knowig that you are the 60 billionth person is irrelevant.

Yes, I agree entirely. But, if you agree with the self-sampling assumption, then these problems become a lot simpler to think about, since one can always come up with a single-urn model in which all observers in the past and future history of the universe/multiverse are individual balls in an urn, and the self-sampling assumption corresponds to picking a ball randomly from this urn. Here are the three cases I think are worth looking at:

1. Only one civilization in entire cosmos, 95% chance of 100 trillion and 5% chance of 100 billion:

Obviously this would correspond to an urn which someone filled with either 100 trillion or 100 billion numbered balls, choosing how many randomly, with a 95% chance of 100 trillion balls and a 5% chance of 100 billion balls. If you pick a ball randomly from this urn and find its number is 100 billion or less, you should now bet there’s a 98% chance the urn only has 100 billion balls, by Bayesian reasoning.

2. Many civilizations in the cosmos, with a known frequency of 95% of all worlds having 100 trillion individuals and 5% having 100 billion individuals:

For simplicity’s sake, let’s think of this as 20 worlds, 1 of which has 100 billion and the rest have 100 trillion. In terms of the urn model, imagine filling an urn with 1,900,100,000,000,000 balls, each labeled by which world it came from (‘world 12’) and a birth rank for an individual on that world. Here it is easy to see that if you draw a ball with a birth rank under 100 billion, that tells you nothing new about the probability that the world on that ball will have 100 billion vs. 100 trillion individual balls, since every single world has exactly one ball with that number on it.

3. Many civilizations in the cosmos, but you don’t know the frequency of 100 trillion vs. 100 billion, 5% is just your best subjective estimate:

To make it possible to analyze this one mathematically, let’s say that although we don’t know the probability of one world vs. another, we do know the meta-probability of different possible probability distributions. Say we know God created exactly 20 worlds, and that he flipped a coin before doing so, saying "heads, all 20 will have 100 trillion individuals, tails, 18 will have 100 trillion and 2 will have 100 billion." So there’s a 50% chance that 0% have 100 billion and a 50% chance that 10% have 100 billion, which leads us to estimate a 5% chance that a randomly-selected world will have 100 billion individuals.

Again, suppose we think of this in terms of a single urn in which all individuals in the cosmos are represented by a ball with their birth rank and the number of the world (1-20) that they come from. The prior probability that the world# on the ball we pick will be one with only 100 billion balls is not 5%, because there are more balls for worlds with 100 trillion individuals, so we’re more likely to pick a ball from such a world--it should actually be (prob. of 2 worlds with 100 billion)(prob. of picking a ball from one of those worlds) + (prob. of 0 worlds with 100 billion)(prob. of picking a ball from one of those worlds) = (0.5)(0.000111…) + (0.5)(0) = 0.0000555. For our Bayesian analysis, this will represents the prior probability of getting a ball belonging to a world with only 100 billion balls--imagine picking a ball, then covering up its birth rank# and looking only at its world#, then being asked the probability that that world# has 100 billion balls going with it instead of 100 trillion.

Suppose we now look at the "birth rank" number on the ball, and see that it is 100 billion or lower. If all 20 worlds had 100 trillion balls, there would be 2*10^15 balls total, and 2*10^12 balls numbered 100 billion or lower, so the probability of getting a ball with rank# 100 billion or lower would be 1 in 1000. But if 18 had 100 trillion and 2 had 100 billion, there would only be 1.8002*10^15 balls total, and there’d still be 2*10^12 balls numbered 100 billion or lower, so the the probability of getting a ball 100 billion or lower would now be 1 in 900.1 (or 10/9001 if you prefer). Using the Bayesian formula, this means that if we do indeed pick a ball with rank 100 billion or lower, our initial estimate of 0.00555% for the probability that the world# of the ball we picked has 100 billion should be modified to about 5.26%. This is a pretty big difference, and it’s ultimately a consequence of the fact that our "prior probability" was not based on certain knowledge about the freqencies of the two types of worlds, but was only our best guess about the frequencies, which could be modified in light of new information (like our own birth rank).