[ExI] kepler study says 8.8e9 earthlike planets

Anders Sandberg anders at aleph.se
Tue Nov 5 17:37:27 UTC 2013


On 2013-11-05 10:40, Adrian Tymes wrote:
> That's the source of your confusion: the illusion of convenient 
> numbers.  The actual percentages are probably unwieldy small fractions 
> of a single percent. Nature doesn't care that we decimalize things.

Bah. See below.

> Consider the exact fraction you would need, for Earth to be the only 
> one.  The reason it feels wrong is because it's an inconvenient fraction.
I have exactly one cup of coffee in front of me. What probability need I 
assume for coffee cups to make it the only one? Clearly, if nobody else 
has a coffee cup, it needs to be way less than one in 7 billion! Wow, 
what a rare coffee cup I have!

Sorry, this is not how it works.

Let's do it right then:

http://aleph0.clarku.edu/~djoyce/ma218/bayes2.pdf 
<http://aleph0.clarku.edu/%7Edjoyce/ma218/bayes2.pdf>
http://www.stat.tamu.edu/~fliang/STAT605/lect01.pdf 
<http://www.stat.tamu.edu/%7Efliang/STAT605/lect01.pdf>
http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture7.pdf 
<http://www.cs.berkeley.edu/%7Ejordan/courses/260-spring10/lectures/lecture7.pdf>

We observe ourselves to be on Earth. What does that do to the 
probability of biospheres being *possible* on exactly Earth-like 
planets? Obviously it sets it to 1. What does it do to the probability p 
of life on similar planets? This is equivalent to doing a Bernouilli 
trial and getting one success. If you start with a uniform prior, then 
the resulting posterior probabability distribution for the real 
probability is now f(p)=2p - a triangular distribution with maximum at p=1.

If we instead use an uninformative Jeffrey prior for a Bernouilli trial, 
P(p) = 1/[pi sqrt(p(1-p))] - a lot of the mass is really close to 0 or 
1, quite inconvenient. In this case the posterior is proportional to 
p/sqrt(p(1-p)). Again most of the probability mass is close to p=1.

If we enlarge the class to planets in or near the life zone, we have one 
success and two failures in the solar system. In this case we get a beta 
distribution as posterior, P(p)=p(1-p)^2/B(2,3) for the uniform prior - 
a softer bulge peaking at p=1/3. Multiplying with a Jeffreys prior 
shifts the peak down a little bit, but not by much.

Now repeat the process with the other planet classes. We do not have any 
known examples, so it will just be priors going into the estimate. The 
expected number of biospheres will be E(sum_i p_i N_i)=sum_i E(p_i N_i) 
where p_i is the probability for class i, N_i the number of planets in 
class i. The expectation for both uniform or Jeffries priors is N/2 - 
far, far more than 1% (since each category has mean p=1/2).

So the rational thing is to expect *lots* of biospheres. Which is of 
course not good news, since that makes a future Great Filter more likely.

-- 
Dr Anders Sandberg
Future of Humanity Institute
Oxford Martin School
Oxford University

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