True. It is not logically implied from him being right that he had good reason to believe he was right. However, I think it is very strong evidence. Fair warning: I am very new to using Bayes’ Theorem, so please make sure to be highly critical of my math, and tell me what, if anything, I’m doing wrong.
First, we must assess the prior probability of Einstein having sufficient evidence, given that he thought he was correct. How often do modern scientists come up with theories that are quickly falsified? Let’s be pessimistic and assign 0.001 prior probability for Einstein making his claim with sufficient evidence. That is, only 1 in 1000 credible scientists who publish theories come up with theories that aren’t easily falsified.
What is the probability of him being correct if he had sufficient evidence? Well, if we say that having sufficient evidence means having evidence such that your prediction has P>0.95, then, if someone has sufficient evidence, their prediction must have P>0.95. Let’s assign a probability of 0.95.
What is the probability of him being correct if he had insufficient evidence? To be strictly logical about this, we would need to take this probability as 0.95 as well, to avoid a false dichotomy. It is not true that either Einstein had p=0.95 worth of evidence or he had no evidence at all. If we say that he necessarily has p > 0.95 given that he has sufficient evidence, we’d have to say that anything under p=0.95 is insufficient evidence; in which case, to be pessimistic, we’d have to assign the probability of him being correct given insufficient evidence to be infinitesimally less than 0.95. This would result in a likelihood ratio of approximately 1. However, this is only the case if we view “insufficient evidence” and “sufficient evidence” to be distinguished by a sharp point on the real number line. This would contradict common sense; we don’t say that p=95 is sufficient but p=94999999 is insufficient. It’s a gradient. So we should choose a number that is definitely insufficient evidence. What you claimed was that “He could have been a lucky draw from the same process”, so let’s go with sufficient evidence being p>0.95 and insufficient being no evidence at all: a guess. With no evidence the probability would be 1/n, n being the number of competing hypotheses. Let’s go with Eliezer’s number of 1⁄100,000,000.
Plugging these numbers into Bayes’ Theorem, we get a posterior probability of roughly 0.999 that Einstein had sufficient evidence to support his belief. Note that this number is larger than 0.95, our previously assumed standard for sufficient evidence. If we instead use 0.99 as our standard, Bayes’ Theorem spits out a posterior probability of roughly 0.999 regardless.
In either case, Einstein being correct about his theory gives us more than enough evidence to conclude that he had sufficient evidence to make the claim.
True. It is not logically implied from him being right that he had good reason to believe he was right. However, I think it is very strong evidence. Fair warning: I am very new to using Bayes’ Theorem, so please make sure to be highly critical of my math, and tell me what, if anything, I’m doing wrong.
First, we must assess the prior probability of Einstein having sufficient evidence, given that he thought he was correct. How often do modern scientists come up with theories that are quickly falsified? Let’s be pessimistic and assign 0.001 prior probability for Einstein making his claim with sufficient evidence. That is, only 1 in 1000 credible scientists who publish theories come up with theories that aren’t easily falsified.
What is the probability of him being correct if he had sufficient evidence? Well, if we say that having sufficient evidence means having evidence such that your prediction has P>0.95, then, if someone has sufficient evidence, their prediction must have P>0.95. Let’s assign a probability of 0.95.
What is the probability of him being correct if he had insufficient evidence? To be strictly logical about this, we would need to take this probability as 0.95 as well, to avoid a false dichotomy. It is not true that either Einstein had p=0.95 worth of evidence or he had no evidence at all. If we say that he necessarily has p > 0.95 given that he has sufficient evidence, we’d have to say that anything under p=0.95 is insufficient evidence; in which case, to be pessimistic, we’d have to assign the probability of him being correct given insufficient evidence to be infinitesimally less than 0.95. This would result in a likelihood ratio of approximately 1. However, this is only the case if we view “insufficient evidence” and “sufficient evidence” to be distinguished by a sharp point on the real number line. This would contradict common sense; we don’t say that p=95 is sufficient but p=94999999 is insufficient. It’s a gradient. So we should choose a number that is definitely insufficient evidence. What you claimed was that “He could have been a lucky draw from the same process”, so let’s go with sufficient evidence being p>0.95 and insufficient being no evidence at all: a guess. With no evidence the probability would be 1/n, n being the number of competing hypotheses. Let’s go with Eliezer’s number of 1⁄100,000,000.
Plugging these numbers into Bayes’ Theorem, we get a posterior probability of roughly 0.999 that Einstein had sufficient evidence to support his belief. Note that this number is larger than 0.95, our previously assumed standard for sufficient evidence. If we instead use 0.99 as our standard, Bayes’ Theorem spits out a posterior probability of roughly 0.999 regardless.
In either case, Einstein being correct about his theory gives us more than enough evidence to conclude that he had sufficient evidence to make the claim.