With many thanks to Damon Binder, and the spirited conversations that lead to this post, and to Anders Sandberg.

People often think that the self-indication assumption (SIA) implies a huge number of alien species, millions of times more than otherwise. Thought experiments like the presumptuous philosopher seem to suggest this.

But here I'll show that, in many cases, updating on SIA doesn't change the expected number of alien species much. It all depends on the prior, and there are many reasonable priors for which the SIA update does nothing more than double the probability of life in the universe^[1].

This can be the case even if the prior says that life is very unlikely! We can have a situation where we are astounded, flabbergasted, and disbelieving about our own existence - "how could we exist, how can this beeeeee?!?!?!?" - and still not update much - "well, life is still pretty unlikely elsewhere, I suppose".

In the one situation where we have an empirical distribution, the "Dissolving the Fermi Paradox" paper, the effect of the SIA anthropics update is to multiply the expected civilization per planet by seven. Not seven orders of magnitude - just seven.

The formula

Let $ρ \in [0, 1]$ be the probability of advanced space-faring life evolving on a given planet; for the moment, ignore issues of life expanding to other planets from their one point of origin. Let $f$ be the prior distribution of $ρ$ , with mean $μ$ and variance $σ^{2}$ . This means that, if we visit another planet, our probability of finding life is $μ$ .

On this planet, we exist^[2]. Then if we update on our existence we get a new distribution $f^{'}$ ; this distribution will have mean $μ^{'}$ :

$μ^{'} = μ (1 + \frac{σ^{2}}{μ^{2}}) .$

To see a proof of this result, look at this footnote^[3].

Define $M_{μ, σ^{2}} = 1 + σ^{2} / μ^{2}$ to be this multiplicative factor between $μ$ and $μ^{'}$ ; we'll show that there are many reasonable situations where $M_{μ, σ^{2}}$ is surprisingly low: think $2$ to $100$ , rather than in the millions or billions.

Beta distributions I

Let's start with the most uninformative prior of all: a uniform prior over $[0, 1]$ . The expectation of $ρ$ is $\int_{0}^{1} ρ d ρ = 1 / 2$ , so, without any other information, we expect a planet to have life with $50 %$ probability. The variance is $σ^{2} = 1 / 12$ .

Thus if we update on our existence on Earth, we get the posterior $f^{'} (ρ) = 2 ρ$ ; the mean of this is $2 / 3$ (either direct calculation or using $M_{1 / 2, 1 / 12} = 1 + 4 / 12 = 4 / 3$ ).

Even though this change in expectation is multiplicativel...

Practical Guide to Anthropics