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Dec 20, 2016

In addition to taking CS 294-115 this semester , I also took STAT 210A, Theoretical Statistics. Here’s a description of the class from the Berkeley Statistics Graduate Student Association (SGSA):

All of the core courses require a large commitment of time, particularly with the homework. Taking more than two of them per semester (eight units) is a Herculean task.

210A contains essential material for the understanding of statistics. 210B contains more specialized material. The 210 courses are less mathematically rigorous than the 205 courses. 210A is easier than 205A; 210B is very difficult conceptually, though in practice it’s easy to perform adequately. Homework is time-consuming.

210A: The frequentist approach to statistics with comparison to Bayesian and decision theory alternatives, estimation, model assessment, testing, confidence regions, some asymptotic theory.

And on our course website we get the following description:

Stat 210A is Berkeley’s introductory Ph.D.-level course on theoretical statistics. It is a fast-paced and demanding course intended to prepare students for research careers in statistics.

Both of these descriptions are highly accurate. I saw these before the semester, so I was already prepared to spend many hours on this course. Now that it’s over, I think I spent about 20 hours a week on it. I thought this was a lot of time, but judging from the SGSA, maybe even the statistics PhD students have to dedicate lots of time to this course?

Our instructor was Professor William Fithian, who is currently in his second year as a statistics faculty member. This course is the first of two theoretical statistics courses offered each year aimed at statistics PhD students. However, I am not even sure if statistics PhD students were in the majority for our class. At the start of the semester, Professor Fithian asked us which fields we were studying, and a lot of people raised their hands when he asked “EECS”. Other departments represented in 210A included mathematics and engineering fields. Professor Fithian also asked if there was anyone studying the humanities or social sciences, but it looks like there were no such students. I wouldn’t expect any; I don’t know why 210A would be remotely useful for them. Maybe I can see a case for a statistically-minded political science student, but it seems like 215A and 215B would be far superior choices.

The number of students wasn’t too large, at least compared to the crowd-fest that’s drowning the EECS department. Right now I see 41 students listed on the bCourses website, and this is generally accurate since the students who drop classes usually drop bCourses as well.

It’s important to realize that even though some descriptions of this course say “introduction” (e.g., see Professor Fithian’s comments above), any student who majored in statistics for undergrad, or who studied various related concepts (computer science and mathematics, for instance) will have encountered the material in 210A before at some level. In my opinion, a succinct description of this course’s purpose is probably: “provide a broad review of statistical concepts but with lots of mathematical rigor , while filling in any minor gaps.”

Similarly, the function det ( A ) 1 / m is concave and the corresponding maximization problem can be phrased in a similar way using a geometric mean bound from Sec. 4.2.4 (Geometric mean) .

We next consider a non-square matrix A R m × p . Assume p m and denote the singular values of A by

σ 1 ( A ) σ 2 ( A ) σ p ( A ) 0.

The singular values are connected to the eigenvalues of A T A via

(6.13) σ i ( A ) = λ i ( A T A ) ,

and if A is square and symmetric then σ i = | λ i | . We show next how to optimize several functions of the singular values.

Largest singular value

The epigraph σ 1 ( A ) t can be characterized using (6.13) as

A T A t 2 I

which from Schur’s lemma is equivalent to

(6.14) FootThrills Plaza Black Leather wFtzg8
[ t I A A T t I ] 0.

The largest singular value σ 1 ( A ) is also called the spectral norm or the 2 -norm of A , A 2 := σ 1 ( A ) .

Processing math: 48%
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A measurable cardinal κ is an uncountable Bamboo Womens Almond Toe Ballet Flat Black nubuck HQtZryWa
such that it is possible to "measure" the subsets of κ using a 2-valued Nike Golf Air Zoom Attack Shoes Gray/White 7 M US 4h9rxtpK
on the powerset of κ , P ( κ ) . There exists several other equivalent definitions: For example, κ can also be the critical point of a nontrivial Shenn Mens Driving Car Slip On Casual Penny Suede Leather Loafers Shoes Navy hXvLLBpL
j : V M .

Every measurable is a large cardinal, i.e. V κ satisfies ZFC , therefore ZFC cannot prove the existence of a measurable cardinal. In fact κ is inaccessible , the κ th inacessible, the κ th weakly compact cardinal, the κ th Ramsey , and similarly bears most of the large cardinal properties under Ramsey-ness. It is notable that every measurable has the mentioned properties in ZFC , but in ZF they may not (but their existence remains consistency-wise much stronger than existence of cardinals with those properties), in fact under the AllhqFashion Womens Round Closed Toe Low Heels Pull On Assorted Color PumpsShoes Red lH3DTfy
, the first two uncountable cardinals, 1 and 2 , are both measurable.

Measurable cardinals were introduced by Stanislaw Ulam in 1930.

Contents

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]

The following definitions are equivalent for every uncountable cardinal κ :

The equivalence between the first two definition is due to the fact that if μ is a 2-valued measure on κ , then U = { X κ | μ ( X ) = 1 } is a nonprincipal ultrafilter (since μ is 2-valued) and is also σ -complete because of μ 's σ -additivity. Similarly, if U is a σ -complete nonprincipal ultrafilter on κ , then μ : P ( κ ) [ 0 , 1 ] defined by μ ( X ) = 1 whenever X U , μ ( X ) = 0 otherwise is a 2-valued measure on κ .

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