Your argument assumes a uniform prior, but a Gaussian prior is more realistic in this case. In practice, IQ scores are distributed normally, so it’s more likely that someone with a high SAT score comes from a more common IQ range than from a very high outlier. For example, say the median rationalist has an SAT score of +2 SD (chosen for ease of computation), and the SAT-IQ correlation is 0.80. The IQ most likely to produce an SAT of +2 SD is 137.5 (+2.5 SD). However, IQs of 137.5 are rare (99.4%-ile). While lower IQs are less likely to achieve such a high SAT score, there are more people in the lower IQ ranges, making it more probable that someone with a +2 SD SAT score falls into a lower IQ bracket.
This shift between the MLE (Maximum Likelihood Estimate) and MAP (Maximum A Posteriori) estimates is illustrated in the graph, where the MLE estimate would be +2.5 SD, but the MAP estimate, accounting for the Gaussian prior, is closer to +1.6 SD, as expected. (You may also be interested in my reply to faul_sname’s comment.)
This doesn’t seem wrong to me so I’m now confused again what the correct analysis is. It would come out the same way if we assume rationalists are selected on g right?
Is a Gaussian prior correct though? I feel like it might be double-counting evidence somehow.
Your argument assumes a uniform prior, but a Gaussian prior is more realistic in this case. In practice, IQ scores are distributed normally, so it’s more likely that someone with a high SAT score comes from a more common IQ range than from a very high outlier. For example, say the median rationalist has an SAT score of +2 SD (chosen for ease of computation), and the SAT-IQ correlation is 0.80. The IQ most likely to produce an SAT of +2 SD is 137.5 (+2.5 SD). However, IQs of 137.5 are rare (99.4%-ile). While lower IQs are less likely to achieve such a high SAT score, there are more people in the lower IQ ranges, making it more probable that someone with a +2 SD SAT score falls into a lower IQ bracket.
This shift between the MLE (Maximum Likelihood Estimate) and MAP (Maximum A Posteriori) estimates is illustrated in the graph, where the MLE estimate would be +2.5 SD, but the MAP estimate, accounting for the Gaussian prior, is closer to +1.6 SD, as expected. (You may also be interested in my reply to faul_sname’s comment.)
This doesn’t seem wrong to me so I’m now confused again what the correct analysis is. It would come out the same way if we assume rationalists are selected on g right?
Is a Gaussian prior correct though? I feel like it might be double-counting evidence somehow.