Comment by mitthrowaway2
3 years ago
> What is the Bayesian prior on "After receiving the COVID vaccination, patient died when they were struck by lightning?"
Mathematically not difficult to deal with, which is why you intuitively know the answer already, because it's approximately the same way your brain works naturally. The chance of dying when you are struck by lightning is about 10%,[1] which completely overwhelms the chance of the vaccine coincidentally killing you at the same moment by several orders of magnitude; while you could use the this datapoint if you want, Bayesian analysis will still confirm that it has approximately zero evidence supporting (or refuting) that the vaccine causes death, so Bayes' theorem effectively drops that patient out of the analysis. Which is the same thing you'd probably have done intuitively, unless some kind of formal controlled-test study rule prevented you from doing what would normally be the correct procedure. (Or worse, forced you to subject an unvaccinated control-group patient to a lightning strike, just to maintain control...?)
But, since your patient died from the lightning strike rather than surviving it, there is a small signal present that points towards "the vaccine increased the chance of the lightning strike being fatal", which is a postulate that deserves a high initial skepticism until someone can propose a mechanism. But if you notice that multiple people die from lightning strikes after getting vaccinated, such that their lightning-strike survival rate is below the unvaccinated population's, you can start to update on that, and if it keeps happening, it becomes worth your time to investigate a potential mechanistic explanation. Like, the vaccination causing people to be less sweaty somehow.
[1] https://www.britannica.com/one-good-fact/what-are-the-odds-o...
... or the unvaccinated population tends to cluster, for sociopolitical reasons, in a region of the country, where lightning strikes are unusually powerful. IRL confounding factors abound.
Sweatiness is a good example of the sort of thing that would be hard to determine by comparing to past sample groups, since it's a rarely recorded piece of data.
Sure, so you add up those terms in the usual way, which you can do by straightforward factorization and summation, until your sum has diminished to probabilities small enough to round off.
This mathematical procedure is usually called "applied common sense", and if you are attempting to argue that it's impossible, you would in the process be proving that nobody can ever learn anything about anything.
You're still assuming that we know all the terms. We use randomized trials with controls improving out medicines because we almost never have enough information to compare two situationally different groups. In a controlled randomized trial, we can declare what we're testing and what we're measuring up front. The data is just too messy, otherwise. Too many hidden variables and too many possible alternate explanations that weren't measured.