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IlyaShpitser comments on Starting University Advice Repository - Less Wrong Discussion
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I think you are wrong about this.
I have no idea whether to downvote. On the one hand, you don't explain why. On the other hand, neither does Lumifer.
But:
I have never in my career needed proofs, and never expect to; I've needed linear algebra on multiple occasions, and expect to many times in the future. I work as a programmer, and have been programming for around twenty years now. I have developed something that twenty years ago might have been called AI. I have managed dozens of projects, and worked on dozens more.
So, where am I going? Proofs are an absolutely worthless skill to have in terms of "getting a bachelor's and going and getting a job". There -might- be a job where you need to write proofs, but if there is, I haven't seen it.
What did you develop 20 years ago?
You should know I ignore karma, btw.
Think about the form of the statement you are making: "I don't know X, and it doesn't seem like I need X." Well, how do you know you don't? You have to compare current world to a counterfactual world where you did know X. How do you know you wouldn't be vastly better off? See also: "I don't need this fancy book learnin' I am doing fine in life."
That's not the statement I am making. I do know X; proofs were required coursework for every CS major at the educational facility I attended. I've never needed it.
Which specific work-related (meta) skill do you think doing proofs develops? It's not going to raise anyone's IQ, I don't see why it would be particularly effective at improving, say, the ability to focus or critical thinking or something like that.
I don't care about IQ, I think it's a fairly uninformative number. Doing proofs eventually gives you a nebulous thing called "mathematical sophistication" (what I sometimes call "metal struts in your brain") that I think helps enormously for adapting to and solving novel technical problems.
When Heinlein said "specialization is for insects" I think he was making a similar point about metaskills.
I don't mean IQ as a number, I mean the underlying g.
And people who graduate college and start working neither do, nor are expected to "solve novel technical problems". The closest to that are programmers who do have to solve problems daily, but for them courses in e.g. data structures or just experience with radically different languages will develop much more useful intuitions than "mathematical sophistication".
If you are going to become a mathematician or a logician, by all means go study proofs. Otherwise I don't think they justify the opportunity costs.
Aren't you suggesting specializing in a particular metaskill?
I think g is sort of a mathematical artifact, not a real thing (but don't really feel like getting into a big thing about this). Factor analysis doesn't tell people what they think it does.
The first principal component of scores on various tests is, of course, a mathematical artifact. But it's not the real thing, it's just an estimate, a finger pointing at the real thing.
I agree that people can be both stupid and smart in very different ways, but at a certain -- and useful! -- level of aggregation, there are generally smart people and generally stupid people. There is a lot of variation around that axis, but I think the axis exists. I'm not arguing that everything should be projected into that one-dimensional space and reduced to a scalar.
Here is how this game works. We have a bunch of observed variables X, and a smaller set of hidden variables Z.
We assume a particular model for the joint distribution p(X,Z). We then think about various facts about this distribution (for example eigenvalues of the covariance matrix). We then try to conclude a causal factor from these facts. This is where the error is. You can't conclude causality that way.
Yes, you can. You can conclude that some causal factor exists. You then define g to be that causal factor.
No you can't conclude that. I am glad we had this chat.
So you're saying that the fact that all these traits are correlated is a complete coincidence?
What if there are several such causal factors?
The correlation studies that lead to defining IQ suggested that there is a single one, or if there multiple, they themselves are strongly correlated with each other.
I know how the game works, I've paged through the Pearl book. But here, in this case, I don't care much about causality. I can observe the existence of stupid people and smart people (and somewhat-stupid, and middle-of-the-road, and a bit smart, etc.). I can roughly rank them on the smart - stupid axis. That axis won't capture all the diversity and the variation, but it will capture some. Whether what it captures is sufficient depends, of course. It depends on the purpose of the exercise and in some cases that's all you need and in some cases it's entirely inadequate. However in my experience that axis is pretty relevant to a lot of things. It's useful.
Note that here no prediction is involved. I'm not talking about whether estimates of g (IQ, basically) can/will predict your success in life or any similar stuff. That's a different discussion.
???
To the extent that you view g as what it is, I have no problem. But people think g is (a) a real thing and (b) causal. It's not at all clear it is either. "Real things" involved in human intelligence are super complicated and have to do with brain architecture (stuff we really don't understand well). We are miles and miles and miles away from "real things" in this setting.
The game I was describing was how PCA works, not stuff in Pearl's book. The point was PCA is just relying on a model of a joint distribution, and you have to be super careful with assumptions to extract causality from that.
I think of g as, basically, a projection from the high-dimensional space of, let's say, mind capabilities into low dimensions, in this case just a single one. Of course it's an "artifact", and of course you lose information when you do that.
However what I mean by g pointing a finger at the real thing is that this high-dimensional cloud has some structure. Things are correlated (or, more generally, dependent on each other). One way -- a rough, simple way -- to get an estimate of one feature of this structure is to do IQ testing. Because it's so simple and because it's robust and because it can be shown to be correlated to a variety of real-life useful things, IQ scores became popular. They are not the Ultimate Explanation for Everything, but they are better than nothing.
With respect to causality, I would say that the high-dimensional cloud of mind capabilities is the "cause". But it's hard to get a handle on it, for obvious reasons, and our one-scalar simplification of the whole thing might or might not be relevant to the causal relationship we are interested in.
P.S.
PCA actually has deeper problems because it's entirely linear and while that makes it easily tractable, real life, especially its biological bits, is rarely that convenient.
Do you think IQ has to be a causal factor to be a good predictor/be meaningful?
No I do not. I think IQ can be a useful predictor for some things (as good as one number can be, really). But that isn't the story with g, is it? It is claimed to be a causal factor.
If we want to do prediction, let's just get a ton of features and use that, like they do in machine learning. Why fixate on one number?
Also -- we know IQ is not a causal factor, IQ is a result of a test (so it's a consequence, not a cause).
Because it makes sense for many different people to study the same number.
In the last month I talked two times about Gottman. The guy got couples into his lab and observed them for 15 minutes while measuring all sorts of variables. Afterwards he did a mathematical model and found that the model has a 91% success rate in predicting whether newly-wed couples will divorce within 10 years.
The problem? The model is likely overfitted. Instead of using the model he generated in his first study I uses a new model for the next study that's also overfitted. If he would have instead work on developing a Gottman metric, other researcher could research the same metric. Other researcher could see what factors correlate with the Gottman metric.
In the case of IQ, IQ is seen as a robust metric. The EPA did studies to estimate how much IQ point are lost due to Mercury pollution. They priced IQ points. The compared the dollar value of the lost IQ points due to Mercury pollution with the cost for filters that reduce Mercury pollution.
That strong datadriven case allowed the EPA under Obama to take bold steps to reduce Mercury pollution. The Koch brothers didn't make a fuss about but payed for the installation of better filters. From their perspective the statistics were robust enough that it doesn't make sense to fight the EPA in the public sphere on the mercury regulation backed up by data driven argument.
The EPA can only do that because IQ isn't a metric that they invented themselves where someone can claim that the EPA simply did p-hacking to make it's case.