I distinctly remember having points taken off of a physics midterm because I didn't show my work. I think I dropped the exam in the waste basket on the way out of the auditorium.

I've always assumed that the problem is three-fold; generating a formal proof is NP-hard, getting the right answer via shortcuts can include cheating, and the faculty's time is limited. Professors/graders do not have the capacity to rigorously demonstrate to themselves that the steps a student has written down actually pinpoint the unique answer. Without access to the student's mind graders are unable to determine if students cheat or not; being able to memorize and/or reproduce the exact steps of a calculation significantly decrease the likelihood of cheating. Even if graders could do one or both of the previous for a single student, they are not 30x or 100x as smart as their students, making it impractical to repeat the process for every student.

That said, I had some very good mathematics teachers in higher level courses who could force students to think, and one in particular who could encourage/demand novelty from students simply by asking them to solve problems that they hadn't yet learned to solve. I didn't realize the power of the latter approach until later (and at the time everyone complained about exams with a median score well under 50%), but his classes were always my favorite.

Beyond Statistics 101

by JonahS 2 min read26th Jun 2015132 comments

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Is statistics beyond introductory statistics important for general reasoning?

Ideas such as regression to the mean, that correlation does not imply causation and base rate fallacy are very important for reasoning about the world in general. One gets these from a deep understanding of statistics 101, and the basics of the Bayesian statistical paradigm. Up until one year ago, I was under the impression that more advanced statistics is technical elaboration that doesn't offer major additional insights  into thinking about the world in general.

Nothing could be further from the truth: ideas from advanced statistics are essential for reasoning about the world, even on a day-to-day level. In hindsight my prior belief seems very naive – as far as I can tell, my only reason for holding it is that I hadn't heard anyone say otherwise. But I hadn't actually looked advanced statistics to see whether or not my impression was justified :D.

Since then, I've learned some advanced statistics and machine learning, and the ideas that I've learned have radically altered my worldview. The "official" prerequisites for this material are calculus, differential multivariable calculus, and linear algebra. But one doesn't actually need to have detailed knowledge of these to understand ideas from advanced statistics well enough to benefit from them. The problem is pedagogical: I need to figure out how how to communicate them in an accessible way.

Advanced statistics enables one to reach nonobvious conclusions

To give a bird's eye view of the perspective that I've arrived at, in practice, the ideas from "basic" statistics are generally useful primarily for disproving hypotheses. This pushes in the direction of a state of radical agnosticism: the idea that one can't really know anything for sure about lots of important questions. More advanced statistics enables one to become justifiably confident in nonobvious conclusions, often even in the absence of formal evidence coming from the standard scientific practice.

IQ research and PCA as a case study

In the early 20th century, the psychologist and statistician Charles Spearman discovered the the g-factor, which is what IQ tests are designed to measure. The g-factor is one of the most powerful constructs that's come out of psychology research. There are many factors that played a role in enabling Bill Gates ability to save perhaps millions of lives, but one of the most salient factors is his IQ being in the top ~1% of his class at Harvard. IQ research helped the Gates Foundation to recognize iodine supplementation as a nutritional intervention that would improve socioeconomic prospects for children in the developing world.

The work of Spearman and his successors on IQ constitute one of the pinnacles of achievement in the social sciences. But while Spearman's discovery of IQ was a great discovery, it wasn't his greatest discovery. His greatest discovery was a discovery about how to do social science research. He pioneered the use of factor analysis, a close relative of principal component analysis (PCA).

The philosophy of dimensionality reduction

PCA is a dimensionality reduction method. Real world data often has the surprising property of "dimensionality reduction":  a small number of latent variables explain a large fraction of the variance in data.

This is related to the effectiveness of Occam's razor: it turns out to be possible to describe a surprisingly large amount of what we see around us in terms of a small number of variables. Only, the variables that explain a lot usually aren't the variables that are immediately visibleinstead they're hidden from us, and in order to model reality, we need to discover them, which is the function that PCA serves. The small number of variables that drive a large fraction of variance in data can be thought of as a sort of "backbone" of the data. That enables one to understand the data at a "macro /  big picture / structural" level.

This is a very long story that will take a long time to flesh out, and doing so is one of my main goals. 

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