<html>
<head>
<meta content="text/html; charset=ISO-8859-1"
http-equiv="Content-Type">
</head>
<body text="#000000" bgcolor="#FFFFFF">
<div class="moz-cite-prefix">On 2013-11-05 10:40, Adrian Tymes
wrote:<br>
</div>
<blockquote
cite="mid:CALAdGNQkRc6fGeDa_do-hZ-qN7T-FXgo820Ap2cYn_pspHSg8Q@mail.gmail.com"
type="cite">
<div dir="ltr">
<div class="gmail_extra">
<div class="gmail_quote">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.<br>
</div>
</div>
</div>
</blockquote>
<br>
Bah. See below. <br>
<br>
<blockquote
cite="mid:CALAdGNQkRc6fGeDa_do-hZ-qN7T-FXgo820Ap2cYn_pspHSg8Q@mail.gmail.com"
type="cite">
<div dir="ltr">
<div class="gmail_extra">
<div class="gmail_quote">
<div>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.<br>
</div>
</div>
</div>
</div>
</blockquote>
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! <br>
<br>
Sorry, this is not how it works. <br>
<br>
Let's do it right then: <br>
<br>
<a href="http://aleph0.clarku.edu/%7Edjoyce/ma218/bayes2.pdf">http://aleph0.clarku.edu/~djoyce/ma218/bayes2.pdf</a><br>
<a href="http://www.stat.tamu.edu/%7Efliang/STAT605/lect01.pdf">http://www.stat.tamu.edu/~fliang/STAT605/lect01.pdf</a><br>
<a
href="http://www.cs.berkeley.edu/%7Ejordan/courses/260-spring10/lectures/lecture7.pdf">http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture7.pdf</a><br>
<br>
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. <br>
<br>
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.<br>
<br>
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. <br>
<br>
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). <br>
<br>
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. <br>
<br>
<pre class="moz-signature" cols="72">--
Dr Anders Sandberg
Future of Humanity Institute
Oxford Martin School
Oxford University
</pre>
</body>
</html>