Suppose that vampires are more likely to play the organ than non-vampires. If you learn that someone is an organ player, does that make them more likely to be a vampire than someone who doesn’t? Let’s assume that you don’t know anything about this person other than the fact that they’re an organ player.
This is completely unrelated to anything in real life. I’m just curious how people here are going to answer the question.
Of course not. Did you just learn Bayes theorem in class today?
Is it also a given that there’s a non-zero probability of vampires existing?
Yes. All four possible combinations (vampire/organ player, vampire/non-organ player, non-vampire/organ player, non-vampire/non-organ player) are represented in the population.
Yes. If a disproportionate number of As are Bs, then a disproportionate number of Bs will be As.
You have a village with a population of 20 people. 10 vampires and 10 non-vampires.
Vampires are 20% likely to play the organ. Non-vampires are 10% likely to play the organ.
This means that 2 vampires and 1 non-vampire play the organ.
Therefore, an organ player is 67% likely to be a vampire even though in the general population, someone is 50% likely to be a vampire.
This works if you change the numbers.
You have a city with 1 000 000 people, 1000 of whom are vampires.
Among non-vampires, 1% play the organ, for a total of 9990 non-vampire organ players.
Among vampires, 2% play the organ, for a total of 20 vampire organ players.
There are a total of 10010 organ players. 1.998% of organ players are vampires, which is nearly twice the proportion of vampires in the general population.
Vampires play bass.
Then the answer is yes. Of course, you have to accept the notion of Bayesian probability in the first place. It’s not intuitively clear that this should be the case: after all, the person in question is either a vampire or not, independent of our own brain state, so why should learning a fact change anything? But on the other hand, Bayesian probability is an extremely useful tool, and seems to match reality, so why not use it?
Of course it is. Do you deny Bayes’s theorem or deny that it is applicable?
Hari, you’re one of 'em mathletes right? My theoretical understanding of that inference is hazy. Could you explain it?
I’d say we don’t have enough data. We don’t know how many organ players there are. We don’t know how many vampires there are.
If there are ten million organ players in the world, and three vampires, and two of them are organ players, then I would argue that the increased probability that any given organ player is a vampire is statistically insignificant.
So I couldn’t answer the poll, because my answer isn’t among the choices.
Sure. Whatever Bayes’ theorem says. 
To intuit, suppose that non-vampire/organ player was not represented at all (i.e. P(organ player | non-vampire) = 0) but all other populations were non-zero. Then if you found out someone was an organ player, you’d be certain that they were a vampire, because there are no non-vampire organists. One would expect that changing P(organ player | non-vampire) to be some tiny value instead wouldn’t suddenly change that certainty back to the worldwide vampire frequency right away.
Phew. I was worried that organ players were real.
I said “no, it’s not a valid inference” precisely because of this. We don’t know what the actual population of vampires/mortals/organists/non-organists are, therefore we can’t make specific predictions about any one individual.
But I’m certainly no mathlete.
This. With the assumption that the percentage of vampires is tiny, it’s not going to make a difference.
Ultrafilter,
Did you mean to ask if that made it more likely that the organ player was a vampire or if that made it more significantly more likely that the organ player was a vampire?
Let’s say there is one vampire in the world. She is an organ player. Therefore 100% of vampires are organ players. Then there are 7,000,000,000 people in the world who are not vampires. Of those, 0.01% play the organ. That means that while vampires are more likely to play the organ than a non vampire, there are 700,000 non vampire organ players in the world. If I hear someone playing the organ, I’m generally going to assume that they are not a vampire.
Yes, but the key term is more likely. The chance of a given non-organ player being a vampire, in this case, is 0, while the chance of an organ player being a vampire is 1/700,000. Therefore, it is safe to assume that an organ player is more likely to be a vampire than a non-organ player, even if the overall probability is still absurdly low.
In other words, it’s more likely for an organ player to be a vampire THAN A NON-ORGAN PLAYER. But it doesn’t follow from that (necessarily) that an organ player is more likely to be a vampire than ANOTHER ORGAN PLAYER.
I don’t know what the word “significant” might mean in this context.
I voted no. Then I re-read the question and realized the answer was yes. But since I’m making the ridiculous assumption that vampires are more likely to be organ players than non-vampires, I’m also going to assume I can be right with both answers. I win. Suck on it organ playing vampires.
Time to compute!
We are given P(O|V) > P(O|-V). This is equivalent to P(OV) > P(O)P(V).
The question is whether P(V|O) > P(V|-O). This is equivalent to P(OV) > P(O)P(V).
Assuming, of course, that all probabilities are non-zero.
Oh snap! Can I change my vote?
Okay, fine. But doesn’t make it a useful inference if you’re trying to discover vampires.