# The probability of extraterrestrial life

Since, the discovery of exoplanets nearly 3 decades ago most astronomers, at least the public facing ones, seem to agree that it is just a matter of time before they find signs of life such as the presence of volatile gases in the atmosphere associated with life like methane or oxygen. I’m an agnostic on the existence of life outside of earth because we don’t have any clue as to how easy or hard it is for life to form. To me, it is equally possible that the visible universe is teeming with life or that we are alone. We simply do not know.

But what would happen if we find life on another planet. How would that change our expected probability for life in the universe? MIT astronomer Sara Seager once made an offhand remark in a podcast that finding another planet with life would make it very likely there were many more. But is this true? Does the existence of another planet with life mean a dramatic increase in the probability of life in the universe. We can find out by doing the calculation.

Suppose you believe that the probability of life on a planet is $f$ (i.e. fraction of planets with life) and this probability is uniform across the universe. Then if you search $n$ planets, the probability for the number of planets with life you will find is given by a Binomial distribution. The probability that there are $x$ planets is given by the expression $P(x | f) = C(x,n) f^x(1-f)^{n-x}$, where $C$ is a factor (the binomial coefficient) such that the sum of $x$ from one to $n$ is 1. By Bayes Theorem, the posterior probability for $f$ (yes, that would be the probability of a probability) is given by

$P(f | x) = \frac{ P(x | f) P(f)}{P(x)}$

where $P(x) = \int_0^1 P(x | f) P(f) df$. As expected, the posterior depends strongly on the prior. A convenient way to express the prior probability is to use a Beta distribution

$P(f |\alpha, \beta) = B(\alpha,\beta)^{-1} f^{\alpha-1} (1-f)^{\beta-1}$ (*)

where $B$ is again a normalization constant (the Beta function). The mean of a beta distribution is given by $E(f) = \alpha/(\alpha + \beta)$ and the variance, which is a measure of uncertainty, is given by $Var(f) = \alpha \beta /(\alpha + \beta)^2 (\alpha + \beta + 1)$. The posterior distribution for $f$ after observing $x$ planets with life out of $n$ will be

$P(f | x) = D f^{\alpha + x -1} (1-f)^{n+\beta - x -1}$

where $D$ is a normalization factor. This is again a Beta distribution. The Beta distribution is called the conjugate prior for the Binomial because it’s form is preserved in the posterior.

Applying Bayes theorem in equation (*), we see that the mean and variance of the posterior become $(\alpha+x)/(\alpha + \beta +n)$ and $(\alpha+x)( \beta+n-x) /(\alpha + \beta + n)^2 (\alpha + \beta + n + 1)$, respectively. Now let’s consider how our priors have updated. Suppose our prior was $\alpha = \beta = 1$, which gives a uniform distribution for $f$ on the range 0 to 1. It has a mean of 1/2 and a variance of 1/12. If we find one planet with life after checking 10,000 planets then our expected $f$ becomes 2/10002 with variance $2\times 10^{-8}$. The observation of a single planet has greatly reduced our uncertainty and we now expect about 1 in 5000 planets to have life. Now what happens if we find no planets. Then, our expected $f$ only drops to 1 in 10000 and the variance is about the same. So, the difference between finding a planet versus not finding a planet only halves our posterior if we had no prior bias. But suppose we are really skeptical and have a prior with $\alpha =0$ and $\beta = 1$ so our expected probability is zero with zero variance. The observation of a single planet increases our posterior to 1 in 10001 with about the same small variance. However, if we find a single planet out of much fewer observations like 100, then our expected probability for life would be even higher but with more uncertainty. In any case, Sara Seager’s intuition is correct – finding a planet would be a game breaker and not finding one shouldn’t really discourage us that much.

## One thought on “The probability of extraterrestrial life”

1. Ishi Crew says:

I grew up as a lonely child–felt all alone in the universe.

I did have many priviledges (since i’m ‘white’ this is called ‘white priviledge’.)

For example, I had a private island-state–half way between Haiti and Hawaii —

according to map the metric distance is ds^2 ~ H^2.

It had its own National Anthem and our state-owned media company similar to RT /NPR/SputnikRadio.

we produced our popular media show which explained the rules on the island –which was called Gilligan’s Island.

Since i was lonely i used to walk around the island to get metrics on our listenership–our market penetration–. and make sure there were no pirate radio stations spreading insurrectionary messages and make sure our national anthem or theme song was #1.

We had pretty much 100% media control.

I also looked to see if there were any other people on the island-state.

I used a simpler formula P(1)=1. P(2 I1) = ? => P (N I 1)

This allowed me to compute the population of my island–i was worried about overpopulation.

Even worse consider if you have P(x,y,z..) in which i become schizophrenic or develop multiple personalities or even diffferent species via parthogenesis and then find more on the island.

What will the limiting distribution be? P(X, 0,0,0..)? or P(x=y=z=..,)

Using the beta distribution seemed arbitrary and too complex so i just looked around.

One time i saw some footprints on the beach–so i knew there was–maybe an ilegal– and followed them all the way around the island and discovered the footprints were my own.

(arthur eddington made a similar discovery on his own ‘ private idaho’
but he had the b-52s to do the sound track for his nation-state).

Then one day the castaways showed up –who named me Robinson Crusoe.
They were very nice –they said ‘what is ours is yours’.

They asked me for the property deeds to my island-state and asked to see the authorities.
since i couldn’t find these they told me to leave–Chrisotopher Columus was their bouncer.
and also told me ‘what is yours is ours’.

So i moved to california and became a museum janitor.

———-

there is a group called 800000 hours whcih spend alot fo time and money calculating how many expoplantes there are

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