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I don't understand what one could mean by the true marginal distribution of a latent quantity that's only defined in terms of its joint distribution with measur

I don't understand what one could mean by the true marginal distribution of a latent quantity that's only defined in terms of its joint distribution with measurements. What difference would it make if we transformed IQ scores to a skewed distribution? To give an uncontroversial example, we use logarithmic scales for earthquakes precisely because the underlying phenomenon is heavy-tailed. It's just for convenience, it doesn't imply anything about the phenomenon being quantified.

runarberg 5y ago
From OP: > The mere fact that the results of these types of tests on large population groups tend to follow a normal distribution is alone good evidence that they're indicative, as in most things diverse large population groups tend to fall on a normal distribution. The claim here is that OP is wrong. Firstly OP is wrong because normal distribution is not a sufficient indicator for some underlying phenomenon. Or as yxwvut puts it: > [...] normal distribution implying some fundamental truth, the same can be said of an iterated guessing game (eg: take the sum of your correct guesses over 100 coin-flips - the population's scores on that game will be normally distributed). And then secondly because intelligence doesn’t even follow a normal curve (only IQ does because it is transformed to). What matter here is not the true distribution of intelligence. I’m not even sure there is one. What matters is that proponents of general intelligence use the normal distribution of IQ as a justification. By arguing that intelligence does not fall on normal distribution but rather on a skewed distribution, I’m simply pulling the rug away from this argument. I admit this is not a strong argument against the existence of general intelligence, since if there exists such a thing, then there is nothing that says such a thing couldn’t exist on a skewed distribution (except psychometricians and proponents of IQ which only accept standard bell curve [I wonder why]). However in this subthread we’re only trying to correct OP’s claim that a normal distribution is indicative for general intelligence.
- duvenaud 5y ago
  Thanks for explaining. It sounds like we agree that "OP is wrong because normal distribution is not a sufficient indicator for some underlying phenomenon". And that we're not sure that there's a "true distribution" of intelligence. But I don't know how you can then argue that "intelligence does not fall on normal distribution but rather on a skewed distribution". Again, how could we choose between calling skewed thing vs its log transform "intelligence"?
  - runarberg 5y ago
    Ahh, I see the confusion now. I actually don’t believe in general intelligence, and I don’t believe intelligence is this one thing that you can measure. So I see how it can be confusing when I go ahead and say: “Intelligence has a skewed distribution.” But when I’m talking about “intelligence” in this context I’m using working within the assumptions of the psychometric. In other words, I forgo my priors and argue on the terms of my opponents. I.e. given that they are right about the existence of general intelligence and that IQ tests are a good and accurate way to measure intelligence, they are still not correct about it being normally distributed. To be precise. When I use the term “intelligence” in this context I’m basically using the same operationally defined term as psychometricians. That is: Intelligence is whatever the IQ tests are measuring. Now as for which distribution to pick for this thing that IQ tests are measuring. Why pick a skewed distribution instead of a normal one? Psychometricians pick the bell curve because they believe it gives this term some validity. I would pick a skewed distribution because that fits our intuition and fits the underlying data. I would call the former justification pseudo-science as you have no reason other then your prior assumptions to do that. Then why do the latter? I would say the only reason to do that is to prove the people doing the former wrong.
    - duvenaud 5y ago
      Thanks for elaborating. But I still don't understand in what sense a skewed distribution "fits the underlying data". I presume that for any given task, we would have to write down a model or likelihood that relates intelligence to the outcome of the task. No matter what transformation of intelligence we use when discussing population statistics, as long as it's monotonic then we can always undo that transformation when we write down a model. Again, regarding earthquakes: Does a logarithmic scale "fit the underlying data better"? I don't think the question even makes sense. It's like asking which font fits the underlying data better. It's an arbitrary choice that doesn't need to affect our predictions or models at all. But you also said that "the only reason to do that is to prove the people doing the former wrong." so maybe we agree on this, too. I think you might be misunderstanding psychometrics if your objection to IQ is the supposed implication that "intelligence is this one thing that you can measure". As far as I understand, psychometricians would agree that intelligence has many facets, and that most of them are hard to measure. I think the claim that they're making is simply that one can summarize estimates of these facets into a single scalar that will be correlated with all of them. Just like GDP correlates with many hard-to-measure and complicated facets of a country's wealth. Would you say the fact that GDP can't measure some important things about a country's wealth, or that it's a complicated thing, implies that there's no such thing as "general wealth"? The same analogy works for e.g. health, strength, or pretty much any property that we have a word for.

runarberg 5y ago

From OP: > The mere fact that the results of these types of tests on large population groups tend to follow a normal distribution is alone good evidence that they're indicative, as in most things diverse large population groups tend to fall on a normal distribution. The claim here is that OP is wrong. Firstly OP is wrong because normal distribution is not a sufficient indicator for some underlying phenomenon. Or as yxwvut puts it: > [...] normal distribution implying some fundamental truth, the same can be said of an iterated guessing game (eg: take the sum of your correct guesses over 100 coin-flips - the population's scores on that game will be normally distributed). And then secondly because intelligence doesn’t even follow a normal curve (only IQ does because it is transformed to). What matter here is not the true distribution of intelligence. I’m not even sure there is one. What matters is that proponents of general intelligence use the normal distribution of IQ as a justification. By arguing that intelligence does not fall on normal distribution but rather on a skewed distribution, I’m simply pulling the rug away from this argument. I admit this is not a strong argument against the existence of general intelligence, since if there exists such a thing, then there is nothing that says such a thing couldn’t exist on a skewed distribution (except psychometricians and proponents of IQ which only accept standard bell curve [I wonder why]). However in this subthread we’re only trying to correct OP’s claim that a normal distribution is indicative for general intelligence.

duvenaud 5y ago
Thanks for explaining. It sounds like we agree that "OP is wrong because normal distribution is not a sufficient indicator for some underlying phenomenon". And that we're not sure that there's a "true distribution" of intelligence. But I don't know how you can then argue that "intelligence does not fall on normal distribution but rather on a skewed distribution". Again, how could we choose between calling skewed thing vs its log transform "intelligence"?
- runarberg 5y ago
  Ahh, I see the confusion now. I actually don’t believe in general intelligence, and I don’t believe intelligence is this one thing that you can measure. So I see how it can be confusing when I go ahead and say: “Intelligence has a skewed distribution.” But when I’m talking about “intelligence” in this context I’m using working within the assumptions of the psychometric. In other words, I forgo my priors and argue on the terms of my opponents. I.e. given that they are right about the existence of general intelligence and that IQ tests are a good and accurate way to measure intelligence, they are still not correct about it being normally distributed. To be precise. When I use the term “intelligence” in this context I’m basically using the same operationally defined term as psychometricians. That is: Intelligence is whatever the IQ tests are measuring. Now as for which distribution to pick for this thing that IQ tests are measuring. Why pick a skewed distribution instead of a normal one? Psychometricians pick the bell curve because they believe it gives this term some validity. I would pick a skewed distribution because that fits our intuition and fits the underlying data. I would call the former justification pseudo-science as you have no reason other then your prior assumptions to do that. Then why do the latter? I would say the only reason to do that is to prove the people doing the former wrong.
  - duvenaud 5y ago
    Thanks for elaborating. But I still don't understand in what sense a skewed distribution "fits the underlying data". I presume that for any given task, we would have to write down a model or likelihood that relates intelligence to the outcome of the task. No matter what transformation of intelligence we use when discussing population statistics, as long as it's monotonic then we can always undo that transformation when we write down a model. Again, regarding earthquakes: Does a logarithmic scale "fit the underlying data better"? I don't think the question even makes sense. It's like asking which font fits the underlying data better. It's an arbitrary choice that doesn't need to affect our predictions or models at all. But you also said that "the only reason to do that is to prove the people doing the former wrong." so maybe we agree on this, too. I think you might be misunderstanding psychometrics if your objection to IQ is the supposed implication that "intelligence is this one thing that you can measure". As far as I understand, psychometricians would agree that intelligence has many facets, and that most of them are hard to measure. I think the claim that they're making is simply that one can summarize estimates of these facets into a single scalar that will be correlated with all of them. Just like GDP correlates with many hard-to-measure and complicated facets of a country's wealth. Would you say the fact that GDP can't measure some important things about a country's wealth, or that it's a complicated thing, implies that there's no such thing as "general wealth"? The same analogy works for e.g. health, strength, or pretty much any property that we have a word for.