Most theorists think they have the right theory but are wrong. So just because Einstein was right, that doesn’t mean he had good reason to believe he was right. He could have been a lucky draw from the same process.
Many theories have been defended on grounds of beauty—and been wrong. Heliocentrism was an elegant theory that worked well and explained many things like the absence of naked-eye precession. Just before Einstein, we can find examples:
According to the vortex atomic theory originally proposed by William
Thomson in 1867, atoms were nothing but vortical structures in the continuous
ether. In this sense the atoms were quasi-material rather than material bodies.
As the ultimate and irreducible quality of nature, the ether could exist without
matter, but matter could not exist without the ether....By the early 1890s the vortex atomic theory had run out of steam and was abandoned by most researchers as a realistic theory of the constitution of matter. It was never unambiguously proved wrong by experiment, but after twenty years of work it degenerated into mathematics, failing to deliver what it promised of physical results. Physicists simply lost confidence in the theory. On the other hand, although the vortex atom was no longer considered a useful concept in explaining physical phenomena, heuristically and as a mental picture it lived on. Wrong as it was, to many British physicists it remained a methodological guiding principle, the ideal of what a future unified theory of matter and ether should look like. According to Michelson, writing in 1903, it “ought to be true even if it is not” (Kragh 2002: 80).
Indeed, I think theorists tend to make mistakes of either deductive or inductive bias. They start out tacitly assuming that reality must be some slightly noisy instantiation of a mathematical theorem … that their favorite equations are logically true and for some mucky reason or another we just observe them as being noisily true.
From the post above:
To assign more than 50% probability to the correct candidate from a pool of 100,000,000 possible hypotheses, you need at least 27 bits of evidence (or thereabouts).
… or you just need to be that one guy who made a wild and unjustified guess about where to assign more than 50 % of the probability (despite not having bits of evidence to support it) and then be lucky.
This is true even if you call your guess a “hunch” or “intuition”.
Only if you make the further assumption that whatever process that generates hunches or intuition must be decision-theoretic. That may not be a bad assumption, but I’m not convinced it’s accurate in human beings. From my own readings about Einstein, I think it’s more likely that he over-asserted the relevance of differential geometry and justified the pursuit of a theory along those lines with what is essentially faith in the mathematics. I don’t think it was a subconscious extension of integrated evidence at all. For every Einstein whose hunch focused on the right general field of mathematics, there were probably dozens or hundreds of other physicists who just thought that burgeoning algebraic topology was the ticket, or perhaps non-standard analysis was the ticket, or perhaps representation theory was the ticket.
I came here with this exact question, and still don’t have a good answer. I feel confident that Eliezer is well aware that lucky guesses exist, and that Eliezer is attempting to communicate something in this chapter, but I remain baffled as to what.
Is the idea that, given our current knowledge that the theory was, in fact, correct, the most plausible explanation is that Einstein already had lots of evidence that this theory was true?
I understand that theory-space is massive, but I can locate all kinds of theories just by rolling dice or flipping coins to generate random bits. I can see how this ‘random thesis generation method’ still requires X number of bits to reach arbitrary theories, but the information required to reach a theory seems orthogonal to the truth. It feels like a stretch to call coin flips “evidence.” I’m guessing that’s what Robin_Hanson2 means by “lucky draw from the same process”; perhaps there were a few bits selected from observation, and a few others that came from lucky coin flips.
Perhaps a better question would be, given a large array of similar scenarios (someone traveling to look at evidence that will possibly refute a theory), how can I use the insight presented in this chapter to constrain anticipation and attempt to perform better than random in guessing which travelers are likely to see the theory violated, and which travelers are not? Or am i thinking of this the wrong way? I remain genuinely confused here, which i hope is a good sign as far as the search for truth :)
My original reading was ‘there was less arrogance in Einstein’s answer than you might think’. After rereading Eliezer’s text and the other comments again today, I cannot tell how much arrogance (regarding rationality) we should assume. I think it is worthwhile to compare Einstein not only to a strong Bayesian:
On the one hand, I agree that a impressive-but-still-human Bayesian would probably have accumulated sufficient evidence at the point of having the worked-out theory that a single experimental result against the theory is not enough to outweigh the evidence. In this case there is little arrogance (if I assume the absolute confidence in “Then I would feel sorry for the good Lord. The theory is correct.” to be rhetoric and not meant literally.)
On the other hand, a random person saying ‘here is my theory that fundamentally alters the way we have to think of our world’ and dismissing a contradicting experiment would be a prime example of arrogance.
Assuming these two cases to be the endpoints of a spectrum, the question becomes where Einstein was located. With special relativity and other significant contributions to physics already at that point in time, I think it is safe to put Einstein into the top tier of physicists. I assume that he did find a strong theory corresponding to his search criteria. But as biases are hard to handle, especially if they concern one’s own assumptions about fundamental principles about our world, there remains the possibility that Einstein did not optimize for correspondence-to-reality for finding general relativity but a heuristic that diverged along the way of finding the theory.
As Einstein had already come up with special relativity (which is related and turned out correct), I tend towards assuming that his assumptions about fundamental principles were on an impressive level, too.
With all this i think it is warranted to take his theory of general relativity very seriously even before the experiment. But Einstein’s confidence is much stronger than that: it seems that he neglects the possibility that some of his fundamental assumptions might be wrong (his confidence in deriving general relativity from these assumptions seems warranted). This means that either he was (to a degree) mistaken in his confidence or that he was on a hard-to-believe level of rationality regarding the question of general relativity. Einstein actually was right, so it is problematic to claim that he was mistaken in his confidence.
After writing this, my conclusion is that i) evidence-gathering for humans might imply that when detecting a signal (finding the theory of general relativity), it’s likely that we have actually accumulated a large pile of evidence, ii) Einstein does seem surprisingly confident, but (i) implies that this could be warranted and it is problematic to criticise correct predictions
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.
Most theorists think they have the right theory but are wrong. So just because Einstein was right, that doesn’t mean he had good reason to believe he was right. He could have been a lucky draw from the same process.
Many theories have been defended on grounds of beauty—and been wrong. Heliocentrism was an elegant theory that worked well and explained many things like the absence of naked-eye precession. Just before Einstein, we can find examples:
--”A Sense of Crisis: Physics in the fin-de-siècle Era”
Indeed, I think theorists tend to make mistakes of either deductive or inductive bias. They start out tacitly assuming that reality must be some slightly noisy instantiation of a mathematical theorem … that their favorite equations are logically true and for some mucky reason or another we just observe them as being noisily true.
From the post above:
… or you just need to be that one guy who made a wild and unjustified guess about where to assign more than 50 % of the probability (despite not having bits of evidence to support it) and then be lucky.
Only if you make the further assumption that whatever process that generates hunches or intuition must be decision-theoretic. That may not be a bad assumption, but I’m not convinced it’s accurate in human beings. From my own readings about Einstein, I think it’s more likely that he over-asserted the relevance of differential geometry and justified the pursuit of a theory along those lines with what is essentially faith in the mathematics. I don’t think it was a subconscious extension of integrated evidence at all. For every Einstein whose hunch focused on the right general field of mathematics, there were probably dozens or hundreds of other physicists who just thought that burgeoning algebraic topology was the ticket, or perhaps non-standard analysis was the ticket, or perhaps representation theory was the ticket.
I came here with this exact question, and still don’t have a good answer. I feel confident that Eliezer is well aware that lucky guesses exist, and that Eliezer is attempting to communicate something in this chapter, but I remain baffled as to what.
Is the idea that, given our current knowledge that the theory was, in fact, correct, the most plausible explanation is that Einstein already had lots of evidence that this theory was true?
I understand that theory-space is massive, but I can locate all kinds of theories just by rolling dice or flipping coins to generate random bits. I can see how this ‘random thesis generation method’ still requires X number of bits to reach arbitrary theories, but the information required to reach a theory seems orthogonal to the truth. It feels like a stretch to call coin flips “evidence.” I’m guessing that’s what Robin_Hanson2 means by “lucky draw from the same process”; perhaps there were a few bits selected from observation, and a few others that came from lucky coin flips.
Perhaps a better question would be, given a large array of similar scenarios (someone traveling to look at evidence that will possibly refute a theory), how can I use the insight presented in this chapter to constrain anticipation and attempt to perform better than random in guessing which travelers are likely to see the theory violated, and which travelers are not? Or am i thinking of this the wrong way? I remain genuinely confused here, which i hope is a good sign as far as the search for truth :)
My original reading was ‘there was less arrogance in Einstein’s answer than you might think’. After rereading Eliezer’s text and the other comments again today, I cannot tell how much arrogance (regarding rationality) we should assume. I think it is worthwhile to compare Einstein not only to a strong Bayesian:
On the one hand, I agree that a impressive-but-still-human Bayesian would probably have accumulated sufficient evidence at the point of having the worked-out theory that a single experimental result against the theory is not enough to outweigh the evidence. In this case there is little arrogance (if I assume the absolute confidence in “Then I would feel sorry for the good Lord. The theory is correct.” to be rhetoric and not meant literally.)
On the other hand, a random person saying ‘here is my theory that fundamentally alters the way we have to think of our world’ and dismissing a contradicting experiment would be a prime example of arrogance.
Assuming these two cases to be the endpoints of a spectrum, the question becomes where Einstein was located. With special relativity and other significant contributions to physics already at that point in time, I think it is safe to put Einstein into the top tier of physicists. I assume that he did find a strong theory corresponding to his search criteria. But as biases are hard to handle, especially if they concern one’s own assumptions about fundamental principles about our world, there remains the possibility that Einstein did not optimize for correspondence-to-reality for finding general relativity but a heuristic that diverged along the way of finding the theory.
As Einstein had already come up with special relativity (which is related and turned out correct), I tend towards assuming that his assumptions about fundamental principles were on an impressive level, too.
With all this i think it is warranted to take his theory of general relativity very seriously even before the experiment. But Einstein’s confidence is much stronger than that: it seems that he neglects the possibility that some of his fundamental assumptions might be wrong (his confidence in deriving general relativity from these assumptions seems warranted). This means that either he was (to a degree) mistaken in his confidence or that he was on a hard-to-believe level of rationality regarding the question of general relativity. Einstein actually was right, so it is problematic to claim that he was mistaken in his confidence.
After writing this, my conclusion is that i) evidence-gathering for humans might imply that when detecting a signal (finding the theory of general relativity), it’s likely that we have actually accumulated a large pile of evidence, ii) Einstein does seem surprisingly confident, but (i) implies that this could be warranted and it is problematic to criticise correct predictions
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.