On updating from dramatic news
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This post is inspired by Astral Codex ten's post Against Learning From Dramatic Events . Let’s start with a short recap.
Let us say that there were only two theories about the danger of current gain-of-function research labs accidentally leaking a pandemic on the world. The ‘safe’ theory, which posits that during the decade the labs have been active there should have been a 10% chance of a leak, and the ‘dangerous’ theory of a 33% chance. Now we get ironclad evidence that Covid was actually caused by a lab leak. How much should our fear of further lab-leak induced pandemics increase?
Scott uses the simple two-theory model to show that our believes in safe/dangerous should increase from 50/50 to 76/24, which leads to a predicted chance of further accidents next decade going from 20% to 27.5%.
This is completely correct with the simple model. And it shows us that we really shouldn’t change our world view too much if we ever find conclusive evidence one way or the other about the Wuhan laboratories.
But what would happen if tried a slightly more complicated model?
My first thought of a model would be one where we have a number of theories, let’s call them Tn, where theory number n predicts n% chance of at least one lab leak. Since we are pretty sure that there haven’t been more than one major leak we should probably only look at models with n<50.1
Let us think up a prior. Thinking up priors is always the best part of going Bayesian. In this case we could perhaps start with the conservative idea of P(Tn)= A/n, where A is just there for normalization. We believe that the safer theories are more likely. Using n=1, 2, 3, … 50 i find a prior chance of at least one leak pr decade to be 11%.
After updating that a lab leak has happened all the theories ends up with the same posterior probability of 1/50, and our believed chance of a leak jumps to 25.5% . Again, not a dramatic increase.
We could of course try other priors. If we let P(Tn)=A/n^2 our chances of a leak goes from 2.7% up to 11%.
For P(Tn)=A/sqrt(n^2) we go from 19% to 30%.
So, unless you started with a belief in lab leaks happening all the time, witnessing one lab leak really doesn’t change that much. And it shows how the simple toy model actually gives a pretty good intuition.
Actually I also tried running this with theories going all the way to 99% chance of a leak. For those you will get a slightly higher increase. But such theories would actually predict that we should have seen many lab-leaked viruses, and so should become even less likely if we find that only one happened during the last decade. They could perhaps be investigated with a poisson distribution.