P(life is common|life on earth)=P(life is common), because knowing that life did evolve on earth can’t give us Bayesian evidence for or against the hypothesis that life is common.
That math is rather obviously wrong. You are so close here—just use Bayes.
Perhaps I should have used an approximately equal to symbol instead of an equals sign, to avoid confusion. And thanks for the detailed writeup. I would agree 100% if you substituted “planet X” for “earth”. Basically, I’m arguing that using ourselves as a data point is a form of the observational selection effect, just like survivorship bias.
Similarly, let’s suppose that we have a less discriminating test, mammography, that still has a 20% rate of false negatives, as in the original case. However, mammography has an 80% rate of false positives. In other words, a patient without breast cancer has an 80% chance of getting a false positive result on her mammography test. If we suppose the same 1% prior probability that a patient presenting herself for screening has breast cancer, what is the chance that a patient with positive mammography has cancer?
Group 1: 100 patients with breast cancer.
Group 2: 9,900 patients without breast cancer.
After mammography* screening:
Group A: 80 patients with breast cancer and a “positive” mammography*.
Group B: 20 patients with breast cancer and a “negative” mammography*.
Group C: 7920 patients without breast cancer and a “positive” mammography*.
Group D: 1980 patients without breast cancer and a “negative” mammography*.
The result works out to 80 / 8,000, or 0.01. This is exactly the same as the 1% prior probability that a patient has breast cancer! A “positive” result on mammography doesn’t change the probability that a woman has breast cancer at all. You can similarly verify that a “negative” mammography also counts for nothing. And in fact it must be this way, because if mammography has an 80% hit rate for patients with breast cancer, and also an 80% rate of false positives for patients without breast cancer, then mammography is completely uncorrelated with breast cancer.
In that example, the reason the posterior probability equals the prior probability is that the “test” isn’t causally linked with the cancer. You have to assume the same the same sort of thing for cases in which you are personally entangled. For example, if I watched my friend survive 100 rounds of solo Russian Roulette, then Baye’s theorem would lead me to believe that there was a high probability that the gun was empty or only had 1 bullet. However, if I myself survived 100 rounds, I couldn’t afterward conclude a low probability, because there would be no conceivable way for me to observe anything but 10 wins. I can’t observe anything if I’m dead.
Does what I’m saying make sense? I’m not sure how else to put it. Are you arguing that Baye’s theorem can still output good data even if you feed it skewed evidence? Or are you arguing that the evidence isn’t actually the result of survivorship bias/observation selection effect?
For example, if I watched my friend survive 100 rounds of solo Russian Roulette, then Baye’s theorem would lead me to believe that there was a high probability that the gun was empty or only had 1 bullet. However, if I myself survived 100 rounds, I couldn’t afterward conclude a low probability, because there would be no conceivable way for me to observe anything but 10 wins. I can’t observe anything if I’m dead.
Obviously you can’t observe anything if you are dead, but that isn’t interesting. What matter is comparing the various hypothesis that could explain the events.
The case where you yourself survive 100 rounds is somewhat special only in that you presumably remember whether you put bullets in or not and thus already know the answer.
Pretend, however that you suddenly wake up with total amensia. There is a gun next to you and a TV then shows a video of you playing 100 rounds of roulette and surviving—but doesn’t show anything before that (where the gun was either loaded or not).
What is the most likely explanation?
the gun was empty in the beginning
the gun had 1 bullet in the beginning
With high odds, option 1 is more likely. This survorship bias/observation selection effect issue you keep bringing up is completely irrelevant when comparing two rival hypothesis that both explain the data!
Here is another, cleaner and simpler example:
Omega rolls a fair die which has N sides. Omega informs you the roll comes up as a ‘2’. Assume Omega is honest. Assume that dice can be either 10 sided or 100 sided, in about the same ratio.
Perhaps I should have used an approximately equal to symbol instead of an equals sign, to avoid confusion. And thanks for the detailed writeup. I would agree 100% if you substituted “planet X” for “earth”. Basically, I’m arguing that using ourselves as a data point is a form of the observational selection effect, just like survivorship bias.
As for the math, I’ll pull an example from An Intuitive Explanation of Bayes’ Theorem:
In that example, the reason the posterior probability equals the prior probability is that the “test” isn’t causally linked with the cancer. You have to assume the same the same sort of thing for cases in which you are personally entangled. For example, if I watched my friend survive 100 rounds of solo Russian Roulette, then Baye’s theorem would lead me to believe that there was a high probability that the gun was empty or only had 1 bullet. However, if I myself survived 100 rounds, I couldn’t afterward conclude a low probability, because there would be no conceivable way for me to observe anything but 10 wins. I can’t observe anything if I’m dead.
Does what I’m saying make sense? I’m not sure how else to put it. Are you arguing that Baye’s theorem can still output good data even if you feed it skewed evidence? Or are you arguing that the evidence isn’t actually the result of survivorship bias/observation selection effect?
Obviously you can’t observe anything if you are dead, but that isn’t interesting. What matter is comparing the various hypothesis that could explain the events.
The case where you yourself survive 100 rounds is somewhat special only in that you presumably remember whether you put bullets in or not and thus already know the answer.
Pretend, however that you suddenly wake up with total amensia. There is a gun next to you and a TV then shows a video of you playing 100 rounds of roulette and surviving—but doesn’t show anything before that (where the gun was either loaded or not).
What is the most likely explanation?
the gun was empty in the beginning
the gun had 1 bullet in the beginning
With high odds, option 1 is more likely. This survorship bias/observation selection effect issue you keep bringing up is completely irrelevant when comparing two rival hypothesis that both explain the data!
Here is another, cleaner and simpler example:
Omega rolls a fair die which has N sides. Omega informs you the roll comes up as a ‘2’. Assume Omega is honest. Assume that dice can be either 10 sided or 100 sided, in about the same ratio.
What is the more likely value of N?
100
10
Here is my solution:
priors: P(N=100) = 1, P(N=10) = 1
P(N=100 | roll(N) = 2) = P(roll(N)=2 | N=100) P(N=100) = 0.01
P(N=10 | roll(N) = 2) = P(roll(N)=2 | N = 10) P(N=10) = 0.1
So N=10 is 10 times more likely than N= 100.