Errr not completely—you have prior and you have experience.
The posterior becomes the next prior when updating again, so we still call it a “prior” even though this is not the same prior as before. Sorry for the confusion. My current prior is my current level of belief/confidence.
Then if you observe miracles you update it to much higher probability
Higher, yes, but (say) ten times almost nothing is still almost nothing. And that’s only if the likelihood ratio for the evidence favors the hypothesis by that much, which it doesn’t.
but you can’t do it if your prior is infinitesimal as now.
That’s right. No finite amount of evidence can overcome an infinitesimal prior.
Your example “miracles” are evidence in favor of miracles existing (because we can hardly expect reports of miracles to be less common if miracles exist) but the likelihood ratio is very close to 1 because false positives (accidents, hallucinations, and hoaxes) are so common. On priors, these explanations are far more likely. That means your “miracle” reports are extremely weak evidence.
I cannot lower my epistemic standards on this, or I would invite in flat-Earthers, UFO-ologists and various other conspiracy theorists, not to mention all the other religions who have similarly dubious paranormal claims. Why should I favor your paranormal claims over theirs? It’s special pleading.
Strong enough evidence can overcome a very low prior, yes. And this doesn’t have to take very many observations.
But more instances do not necessarily stack like that. That can only happen to the degree they are independent sources. For example, suppose you write a dubious claim in a book, then you make nine more copies of the book. Does that make the claim ten times more likely to be true? What if it’s a hundred thousand copies? Did that help?
Of course it doesn’t! You’re re-counting the same evidence. The contribution of the nine books is completely screened off by the first; the new books have no new information.
I think the cases of miracle reports like weeping icons are similarly not independent enough. A thousand weeping icons is barely more evidence than one. It just means that the hoaxers copied each other’s scam.
Furthermore, we already know that some similar instances of miracles were hoaxes. Shouldn’t every new hoax report lower my prior that miracles are real?
The posterior becomes the next prior when updating again, so we still call it a “prior” even though this is not the same prior as before. Sorry for the confusion. My current prior is my current level of belief/confidence.
Higher, yes, but (say) ten times almost nothing is still almost nothing. And that’s only if the likelihood ratio for the evidence favors the hypothesis by that much, which it doesn’t.
That’s right. No finite amount of evidence can overcome an infinitesimal prior.
Your example “miracles” are evidence in favor of miracles existing (because we can hardly expect reports of miracles to be less common if miracles exist) but the likelihood ratio is very close to 1 because false positives (accidents, hallucinations, and hoaxes) are so common. On priors, these explanations are far more likely. That means your “miracle” reports are extremely weak evidence.
I cannot lower my epistemic standards on this, or I would invite in flat-Earthers, UFO-ologists and various other conspiracy theorists, not to mention all the other religions who have similarly dubious paranormal claims. Why should I favor your paranormal claims over theirs? It’s special pleading.
″ but (say) ten times almost nothing is still almost nothing”
Ok, cool. So if your prior will be one millionnth I will need just six miracles :)
Strong enough evidence can overcome a very low prior, yes. And this doesn’t have to take very many observations.
But more instances do not necessarily stack like that. That can only happen to the degree they are independent sources. For example, suppose you write a dubious claim in a book, then you make nine more copies of the book. Does that make the claim ten times more likely to be true? What if it’s a hundred thousand copies? Did that help?
Of course it doesn’t! You’re re-counting the same evidence. The contribution of the nine books is completely screened off by the first; the new books have no new information.
I think the cases of miracle reports like weeping icons are similarly not independent enough. A thousand weeping icons is barely more evidence than one. It just means that the hoaxers copied each other’s scam.
Furthermore, we already know that some similar instances of miracles were hoaxes. Shouldn’t every new hoax report lower my prior that miracles are real?