Bayesian Doomsday Argument

post by DanielLC · 2010-10-17T22:14:17.440Z · LW · GW · Legacy · 16 comments

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First, if you don't already know it, Frequentist Doomsday Argument:

There's some number of total humans. There's a 95% chance that you come after the last 5%. There's been about 60 to 120 billion people so far, so there's a 95% chance that the total will be less than 1.2 to 2.4 trillion.

I've modified it to be Bayesian.

First, find the priors:

Do you think it's possible that the total number of sentients that have ever lived or will ever live is less than a googolplex? I'm not asking if you're certain, or even if you think it's likely. Is it more likely than one in infinity? I think it is too. This means that the prior must be normalizable.

If we take P(T=n) ∝ 1/n, where T is the total number of people, it can't be normalized, as 1/1 + 1/2 + 1/3 + ... is an infinite sum. If it decreases faster, it can at least be normalized. As such, we can use 1/n as an upper limit.

Of course, that's just the limit of the upper tail, so maybe that's not a very good argument. Here's another one:

We're not so much dealing with lives as life-years. Year is a pretty arbitrary measurement, so we'd expect the distribution to be pretty close for the majority of it if we used, say, days instead. This would require the 1/n distribution.

After that,

T = total number of people

U = number you are

P(T=n) ∝ 1/n
U = m
P(U=m|T=n) ∝ 1/n
P(T=n|U=m) = P(U=m|T=n) * P(T=n) / P(U=m)
= (1/n^2) / P(U=m)
P(T>n|U=m) = ∫P(T=n|U=m)dn
= (1/n) / P(U=m)
And to normalize:
P(T>m|U=m) = 1
= (1/m) / P(U=m)
m = 1/P(U=m)
P(T>n|U=m) = (1/n)*m
P(T>n|U=m) = m/n

So, the probability of there being a total of 1 trillion people total if there's been 100 billion so far is 1/10.

There's still a few issues with this. It assumes P(U=m|T=n) ∝ 1/n. This seems like it makes sense. If there's a million people, there's a one-in-a-million chance of being the 268,547th. But if there's also a trillion sentient animals, the chance of being the nth person won't change that much between a million and a billion people. There's a few ways I can amend this.

First: a = number of sentient animals. P(U=m|T=n) ∝ 1/(a+n). This would make the end result P(T>n|U=m) = (m+a)/(n+a).

Second: Just replace every mention of people with sentients.

Third: Take this as a prediction of the number of sentients who aren't humans who have lived so far.

The first would work well if we can find the number of sentient animals without knowing how many humans there will be. Assuming we don't take the time to terreform every planet we come across, this should work okay.

The second would work well if we did tereform every planet we came across.

The third seems a bit wierd. It gives a smaller answer than the other two. It gives a smaller answer than what you'd expect for animals alone. It does this because it combines it for a Doomsday Argument against animals being sentient. You can work that out separately. Just say T is the total number of humans, and U is the total number of animals. Unfortunately, you have to know the total number of humans to work out how many animals are sentient, and vice versa. As such, the combined argument may be more useful. It won't tell you how many of the denizens of planets we colonise will be animals, but I don't think it's actually possible to tell that.

One more thing, you have more information. You have a lifetime of evidence, some of which can be used in these predictions. The lifetime of humanity isn't obvious. We might make it to the heat death of the universe, or we might just kill each other off in a nuclear or biological war in a few decades. We also might be annihilated by a paperclipper somewhere in between. As such, I don't think the evidence that way is very strong.

The evidence for animals is stronger. Emotions aren't exclusively intelligent. It doesn't seem animals would have to be that intelligent to be sentient. Even so, how sure can you really be. This is much more subjective than the doomsday part, and the evidence against their sentience is staggering. I think so anyway, how many animals are there at different levels of intelligence?

Also, there's the priors for total human population so far. I've read estimates vary between 60 and 120 billion. I don't think a factor of two really matters too much for this discussion.

So, what can we use for these priors?

Another issue is that this is for all of space and time, not just Earth.

Consider that you're the mth person (or sentient) from the lineage of a given planet. l(m) is the number of planets with a lineage of at least m people. N is the total number of people ever, n is the number on the average planet, and p is the number of planets.

l(m)/N
=l(m)/(n*p)
=(l(m)/p)/n

l(m)/p is the portion of planets that made it this far. This increases with n, so this weakens my argument, but only to a limited extent. I'm not sure what that is, though. Instinct is that l(m)/p is 50% when m=n, but the mean is not the median. I'd expect a left-skew, which would make l(m)/p much lower than that. Even so, if you placed it at 0.01%, this would mean that it's a thousand times less likely at that value. This argument still takes it down orders of magnitude than what you'd think, so that's not really that significant.

Also, a back-of-the-envolope calculation:

Assume, against all odds, there are a trillion times as many sentient animals as humans, and we happen to be the humans. Also, assume humans only increase their own numbers, and they're at the top percentile for the populations you'd expect. Also, assume 100 billion humans so far.

n = 1,000,000,000,000 * 100,000,000,000 * 100

n = 10^12 * 10^11 * 10^2

n = 10^25

Here's more what I'd expect:

Humanity eventually puts up a satilite to collect solar energy. Once they do one, they might as well do another, until they have a dyson swarm. Assume 1% efficiency. Also, assume humans still use their whole bodies instead of being a brain in a vat. Finally, assume they get fed with 0.1% efficiency. And assume an 80-year lifetime.

n = solar luminosity * 1% / power of a human * 0.1% * lifetime of Sun / lifetime of human

n = 4 * 10^26 Watts * 0.01 / 100 Watts * 0.001 * 5,000,000,000 years / 80 years

n = 2.5 * 10^27

By the way, the value I used for power of a human is after the inefficiencies of digesting.

Even with assumptions that extreme, we couldn't use this planet to it's full potential. Granted, that requires mining pretty much the whole planet, but with a dyson sphere you can do that in a week, or two years with the efficiency I gave.

It actually works out to about 150 tons of Earth per person. How much do you need to get the elements to make a person?

Incidentally, I rewrote the article, so don't be surprised if some of the comments don't make sense.

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