What math is essential to the art of rationality?

Capla

I have started to put together a sort of curriculum for learning the subjects that lend themselves to rationality. It includes things like experimental methodology and cognitive psychology (obviously), along with "support disciplines" like computer science and economics. I think (though maybe I'm wrong) that mathematics is one of the most important things to understand.

Eliezer said in the simple math of everything:

It seems to me that there's a substantial advantage in knowing the drop-dead basic fundamental embarrassingly simple mathematics in as many different subjects as you can manage. Not, necessarily, the high-falutin' complicated damn math that appears in the latest journal articles. Not unless you plan to become a professional in the field. But for people who can read calculus, and sometimes just plain algebra, the drop-dead basic mathematics of a field may not take that long to learn. And it's likely to change your outlook on life more than the math-free popularizations or the highly technical math.

I want to have access to outlook-changing insights. So, what math do I need to know? What are the generally applicable mathematical principles that are most worth learning? The above quote seems to indicate at least calculus, and everyone is a fan of Bayesian statistics (which I know little about).

Secondarily, what are some of the most important of that "drop-dead basic fundamental embarrassingly simple mathematics" from different fields? What fields are mathematically based, other than physics and evolutionary biology, and economics?

What is the most important math for an educated person to be familiar with?

As someone who took an honors calculus class in high school, liked it, and did alright in the class, but who has probably forgotten most of it by now and needs to relearn it, how should I go about learning that math?

Eliezer said in the simple math of everything:

It seems to me that there's a substantial advantage in knowing the drop-dead basic fundamental embarrassingly simple mathematics in as many different subjects as you can manage. Not, necessarily, the high-falutin' complicated damn math that appears in the latest journal articles. Not unless you plan to become a professional in the field. But for people who can read calculus, and sometimes just plain algebra, the drop-dead basic mathematics of a field may not take that long to learn. And it's likely to change your outlook on life more than the math-free popularizations or the highly technical math.

What is the most important math for an educated person to be familiar with?

That's an interesting example, thanks for linking it. I read it carefully, and also some of Robins/Ritov CODA paper:

http://www.biostat.harvard.edu/robins/coda.pdf

and I think I get it. The example is phrased in the language of sampling/missing data, but for those in the audience familiar w/ Pearl, we can rephrase it as a causal inference problem. After all, causal inference is just another type of missing data problem.

We have a treatment A (a drug), and an outcome Y (death). Doctors assign A to some patients, but not others, based on their baseline covariates C. Then some patients die. The resulting data is an observational study, and we want to infer from it the effect of drug on survival, which we can obtain from p(Y | do(A=yes)).

We know in this case that p(Y | do(A=yes)) = sum{C} p(Y | A=yes,C) p(C) (this is just what "adjusting for confounders" means).

If we then had a parametric model for E[Y | A=yes,C], we could just fit that model and average (this is "likelihood based inference.") Larry and Jamie are worried about the (admittedly adversarial) situation where maybe the relationship between Y and A and C is really complicated, and any specific parametric model we might conceivably use will be wrong, while non-parametric methods may have issues due to the curse of dimensionality in moderate samples. But of course the way we specified the problem, we know p(A | C) exactly, because doctors told us the rule by which they assign treatments.

Something like the Horvitz/Thompson estimator which uses this (correct) model only, or other estimators which address issues with the H/T estimator by also using the conditional model for Y, may have better behavior in such settings. But importantly, these methods are exploiting a part of the model we technically do not need (p(A | C) does not appear in the above "adjustment for confounders" expression anywhere), because in this particular setting it happens to be specified exactly, while the parts of the models we do technically need for likelihood based inference to work are really complicated and hard to get right at moderate samples.

But these kinds of estimators are not Bayesian. Of course arguably this entire setting is one Bayesians don't worry about (but maybe they should? These settings do come up).

The CODA paper apparently stimulated some subsequent Bayesian activity, e.g.:

http://www.is.tuebingen.mpg.de/fileadmin/user_upload/files/publications/techreport2007_6326[0%5D.pdf

So, things are working as intended :).

You're welcome for the link, and it's more than repaid by your causal inference restatement of the Robins-Ritov problem.

Of course arguably this entire setting is one Bayesians don't worry about (but maybe they should? These settings do come up).

Yeah, I think this is the heart of the confusion. When you encounter a problem, you can turn the Bayesian crank and it will always do the Right thing, but it won't always do the right thing. What I find disconcerting (as a Bayesian drifting towards frequentism) is that it's not obvious how to assess the adequacy... (read more)

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What math is essential to the art of rationality?

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What math is essential to the art of rationality?

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