I don’t think this math is right. There are two general problems I see with it. Number one is that once you find out this,
Non-heterosexuals are a small minority of the population, but are disproportionally represented among all those who are vaccinated.
your P(infection|vaccine) and P(infection|~vaccine) change to values that make more intuitive sense.
Number two is that I don’t think the percentages you gave in the problem are actually coherent. I don’t believe it’s possible to get the 25% infected and vaccinated and 10% infected and unvaccinated with the numbers provided, no matter what values you use for the rate of heterosexuality in the population, and in the vaccinated and unvaccinated groups. I don’t really want to discuss that in detail, so for the sake of my post here, I’m going to say that 80% of the population is heterosexual, and that 85% of the individuals who have gotten the vaccine are non-heterosexual.
Using these numbers, I calculate that the initial study would find that 26.85% of the vaccinated individuals are infected, and 14.70555...% of the unvaccinated individuals are infected. Note that given just this information, these are the probabilities that a bayesian should assign to the proposition(s), “If I (do not) get the vaccine, I will become infected”. (Unless of course said bayesian managed to pay attention enough to realize that this wasn’t a randomized, controlled experiment, and that therefore the results are highly suspect).
However, once we learn that this survey wasn’t performed using proper, accurate scientific methods, and we gather more data (presumably paying a bit more attention to the methodology), we can calculate a new P(infected|vaccinated) and a new P(infected|unvaccinated) for our child of unknown sexual orientation. As I calculate it, if you believe that your child is heterosexual with 80% confidence (which is the general rate of heterosexuality in the population, in our hypothetical scenario), you calculate that P(infection|vaccinated) = .132, and P(infection|unvaccinated) = .168. So, EDT says to get the vaccine. Alternatively, let’s say you’re only 10% confident your child is heterosexual. In this case, P(infection|vaccinated) is .279, and P(infection|unvaccinated) = .371. Definitely get the vaccine. Say you’re 90% confident your child is heterosexual. Then P(infection|vaccinated) = .111, while P(infection|unvaccinated) = .139. Still, get the vaccine.
Ultimately, using the raw data from the biased study cited initially as your actual confidence level makes about as much sense as applying Laplace’s rule of succession in a case where you know the person drawing the balls is searching through the bag to draw out only the red balls, and concluding that if you draw out a ball from the bag without looking, it will almost certainly be red. It’s simply the wrong way for a bayesian to calculate the probability.
If anything, I think this hypothetical scenario is not so much a refutation of EDT, so much as a demonstration of why proper scientific methodology is important.
I don’t think this math is right. There are two general problems I see with it. Number one is that once you find out this,
your P(infection|vaccine) and P(infection|~vaccine) change to values that make more intuitive sense.
Number two is that I don’t think the percentages you gave in the problem are actually coherent. I don’t believe it’s possible to get the 25% infected and vaccinated and 10% infected and unvaccinated with the numbers provided, no matter what values you use for the rate of heterosexuality in the population, and in the vaccinated and unvaccinated groups. I don’t really want to discuss that in detail, so for the sake of my post here, I’m going to say that 80% of the population is heterosexual, and that 85% of the individuals who have gotten the vaccine are non-heterosexual.
Using these numbers, I calculate that the initial study would find that 26.85% of the vaccinated individuals are infected, and 14.70555...% of the unvaccinated individuals are infected. Note that given just this information, these are the probabilities that a bayesian should assign to the proposition(s), “If I (do not) get the vaccine, I will become infected”. (Unless of course said bayesian managed to pay attention enough to realize that this wasn’t a randomized, controlled experiment, and that therefore the results are highly suspect).
However, once we learn that this survey wasn’t performed using proper, accurate scientific methods, and we gather more data (presumably paying a bit more attention to the methodology), we can calculate a new P(infected|vaccinated) and a new P(infected|unvaccinated) for our child of unknown sexual orientation. As I calculate it, if you believe that your child is heterosexual with 80% confidence (which is the general rate of heterosexuality in the population, in our hypothetical scenario), you calculate that P(infection|vaccinated) = .132, and P(infection|unvaccinated) = .168. So, EDT says to get the vaccine. Alternatively, let’s say you’re only 10% confident your child is heterosexual. In this case, P(infection|vaccinated) is .279, and P(infection|unvaccinated) = .371. Definitely get the vaccine. Say you’re 90% confident your child is heterosexual. Then P(infection|vaccinated) = .111, while P(infection|unvaccinated) = .139. Still, get the vaccine.
Ultimately, using the raw data from the biased study cited initially as your actual confidence level makes about as much sense as applying Laplace’s rule of succession in a case where you know the person drawing the balls is searching through the bag to draw out only the red balls, and concluding that if you draw out a ball from the bag without looking, it will almost certainly be red. It’s simply the wrong way for a bayesian to calculate the probability.
If anything, I think this hypothetical scenario is not so much a refutation of EDT, so much as a demonstration of why proper scientific methodology is important.
Well, you’re right that I did make up the numbers without checking anything. :(
Here’s a version with numbers that work, courtesy of Judea Pearl’s book: http://escholarship.org/uc/item/3s62r0d6