Peter > Status Update

Chapter on hypothesis testing has some good stuff, including a discussion of the contrapositive and of how that notion helps explain why the "null hypothesis" is defined the way if is. The book's definition of Type II error, though, is confusing in its wording. A single poorly-thought-out phrase of math exposition can throw the reader off, to a surprisingly large degree...

Chapter 14, "Confidence In Models," has some clarifying points and some very confusing points. The distinction between precision of a fitted model value (which ignores population variation) and precision of a predicted value (which does not) is sensible enough. But the book's discussion of resampling distributions and simulations is maddeningly unclear.

Two very different types of variation: When you take a statistical sample, you estimate the mean and standard deviation of the population. You also estimate the standard deviation of the mean you just estimated ("standard error"). The standard error of the mean goes to zero as sample size goes to infinity; the standard deviation of the population generally does not.

Having a REALLY hard time letting go. Two random vectors in high-dimensional space are probably close to orthogonal. At some point I want to really convince myself of this. I will try and set it aside and just believe it for right now.

And now the book has got me thinking about spherical coordinates and the angle between two randomly chosen vectors. More gaps to fill in my math knowledge. It takes some effort to keep plowing through the book and setting aside these trains of thought, which I can always revisit when re-reading.

This book may be to statistics what Hughes-Hallett et al is to calculus and Strang is to linear algebra: An excellent, visionary, revolutionary approach for undergrad math ed, but I would hate to teach from it. For students new to the topic, there are too many gaps to fill in.

A very important pair of definitions here: what the author calls "case space" and "variable space." The former is the row space, the latter the column space, in a matrix of data. The rows or "cases" make up the sample to which a model is to be fitted. The columns include data values (e.g., age, height, sex, income) as well as the constant column vector [1,1,...,1] and calculated values (e.g., age times income).

The book is discussing vector spaces, a topic that's old hat to me in some contexts--but my unfamiliarity with "model space" and "error space" in statistics is more-or-less the main reason why I'm reading this book. By the way: I just realized that in 100-dimensional space, the point (0.1, 0.1, ..., 0.1) lies on the unit sphere; that is, its distance from (0,0,...,0) is 1 unit. Those dimensions add up...

Slow going, in part because I keep going on trains of thought as I try to boil everything down to first principles. For example, a simple exposition on sum-of-squares led me to Google "intuitive proof pythagorean theorem" and get lost in that for half an hour. Note to self: read more someday about PT and how rotations preserve Euclidean distance.