Anthropic reasoning 2: the merciful Dr. Killian.

In part 1, I argued that taking your own existence as a datum is coherent and, at least in some situations, necessary. To show that, I gave a simple barebones example of Dr. Killian, who flips a coin and, if it lands tails, kills you in your sleep. If you then wake up you know the coin must have landed heads. What evidence leads you to this conclusion? It's the evidence that you are alive of course! In other words, if the coin didn't land heads you would not be there to even contemplate the question.

At this point you might be thinking that so far this is completely trivial and, if you read part 1, you might be wondering: how does this lead to my claim in the Abiogenesis example that we should think there are more aliens out there than chemistry alone would suggest? I gave a quick intuitive reason in part 1, but to really understand the reason let's consider a slightly less trivial version of the Dr. Killian example.

Life and death experiment 2.

While you are sleeping the unhinged Dr. Killian flips a coin. If it falls heads he does nothing. If it falls tails he almost certainly kills you in your sleep, let's say he lets you live only with probability p = 1%. Later you wake up and find out about this setup, what should you conclude about the coin?

Qualitatively speaking, this is not too different from the previous version, you should conclude that the coin probably fell heads, because otherwise you would probably be dead. This is already pretty good and would allow us to understand the claim in Abiogenesis qualitatively, but we can be more precise. Saying "probably heads" is pretty vague, we want to know how probably. What probability should you assign to heads? 

The answer is 100/101. Why? The answer can be obtained in different ways. One way is with the ensemble method, another is with Bayes' theorem. But let's keep this simple. The basic reason is that if heads you would be alive with probability 100%, and if tails - only 1%. So your observations ("I am alive") are 100 times more likely on heads than on tails. Originally heads and tails are, of course, equally likely, but now, given your observations, you should give a 100 times more credence to heads than tails. So you should assign 100:1 odds to heads vs tails, which translates into the probabilities of 100/101 vs 1/101.

The above was a somewhat hand-wavy, but hopefully reasonably sounding, version of a Bayesian calculation, which, if you are familiar with Bayes' theorem (and you don't have to be for this article), goes like this:

P(H | E) / P(T | E) = P(E | H) * P(H) / [ P(E | T) * P(T) ] = P(E | H) / P(E | T) = 100% / 1% = 100,

where H is heads, T is tails, and E is the evidence that you are alive.

Of course, we used p = 1% for this example but the logic would be exactly the same for any p. We are now very close to be able to reach some very interesting conclusions, including the claim in Abiogenesis. Let's take one more small step and consider another slight modification of the story.

Life and death experiment 3.

Now Dr. Killian has 100 patients, sleeping in separate rooms, he wants to experiment on. He flips a coin and, again, if it falls heads does nothing, and all 100 patients later wake up. If it falls tails then he randomly selects one lucky patient who he would let live and kills the other 99. You are one of the patients. You wake up and find out about the setup. You know you survived but don't know what happened to the patients in the other rooms. What should you conclude about the coin?

This is actually pretty much the same problem as the previous one. Why? Because for you the chance of survival on heads is 100%, and on tails it's 1%, exactly as in the previous problem. So the answer is the same, you should think heads is more likely, specifically the odds of heads vs tails are 100:1.

Abiogenesis revisited.

We are now ready to tackle the example we started with in part 1, which I will summarize here. Suppose, as far as chemistry is concerned, two models about DNA formation are equally plausible, let's call them the Heads model and the Tails model:

• Heads: if this model is true then 100 planets per galaxy develop life,
• Tails: if this model is true then only 1 planet per galaxy develops life.

What should you conclude about which of the two models is correct? This is of course a toy example, especially unrealistic is the stipulation that in any galaxy the number of inhabited planets is either always 1 or always 100. In reality the number would of course vary and depend on the size of the galaxy. But the conclusion of this toy example easily generalizes to more realistic questions.

In part 1 I claimed that even though chemistry alone gives the two possibilities a 50-50 chance, our existence is evidence for Heads, and we should think that most likely Heads is true. We are now more equipped to see why and quantify our conclusion.

Notice that this problem is actually essentially the same as our last iteration of the Dr. Killian story. Let's compare. In the Dr. K story, heads and tails were a priori equally likely, and we had:

• Heads: in this case you are one of 100 existing alive patients
• Tails: in this case you are the only alive patient (in this hospital)

And in Abiogenes we have:

• Heads: in this case we are one of 100 existing civilizations 
• Tails: in this case wee are the only civilization (in this galaxy)

And so the answer here is also the same: you should assign 100:1 to Heads vs Tails, i.e. probabilities 100/101 vs 1/101. Our existence gives us evidence for there being more life in the universe than chemistry alone would suggest!

Objection 1. But there is only one hospital with Dr. K, while there are many galaxies.

A. That doesn't actually change anything, the important part is the ratio: there are 100 times more civilizations on Heads than on Tails, and that's what determines the odds. If in the Dr. K story we had, for example, 700 survivors for Heads and 7 for Tails (instead of 100 and 1), everything would work exactly the same and we would still get 100:1 odds.

Objection 2. If a planet is inhabited it doesn't mean that a civilization has developed on it.

A. That doesn't change anything either for the exact same reason, the important part is the 100 to 1 ratio.

Objection 3. A civilization, unlike Dr. K's patients, is not a conscious thinking agent.

A. I hope it's relatively clear intuitively that this shouldn't make a difference. The reason comes down to the ratio again. If every galaxy has 100 inhabited planets there will on average be 100 times more thinking agents in total than if it has only 1 inhabited planet. In particular, there will on average be 100 times more agents contemplating this question.