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.”

I am not a statistics PhD student nor did I major in statistics during undergrad, which at the time I was there didn’t even offer a statistics major. I took only three statistics courses in college: an introductory to stats course (the easy kind, where we plug in numbers and compute standard deviations), a statistical modeling course (a much easier version than STAT 215A at Berkeley), and an upper-level Bayesian Statistics course taught by Wendy Wang. That third course was enormously helpful, and I’m really starting to appreciate taking that class. The fourth undergrad course that was extremely beneficial was Steven Miller’s Math 341 class (probability), which is also now cross-listed into statistics. I should thank my old professors …

Despite not majoring in statistics, I still knew a lot of concepts that were introduced in this class, such as:

  • Sufficient Statistics
  • Exponential Families
  • Maximum Likelihood
  • Neyman’s Factorization Theorem
  • Jensen’s Inequality and KL Divergences
  • Bootstrapping
  • The Central Limit Theorem

But I didn’t know a lot:

  • Rao-Blackwell Theorem
  • GLRT, Wald, Score, Max tests
  • Convergence in probability vs distribution
  • The Delta Method
  • A lot about hypothesis testing, UMP tests, UMPU tests, etc.

Even for the concepts I already knew, we went into far more detail in this class (e.g. with sufficient statistics) and described the math foundations, so I definitely got a taste as to what concepts appear in statistics research. A good example is the preprint about reranking in exponential families written by the professor and his graduate student Kenneth Hung, who was also the lone TA for the course. Having taken this course, it’s actually possible for me to read the paper and remotely understand what they’re doing.

The syllabus for 210A roughly proceeded in the order of Robert Keener’s textbook, a book which by itself requires significant mathematical maturity to understand. We covered probably more than half of the book. Towards the end of the semester, we discussed more hypothesis testing, which judging from Professor Fithian’s publications (e.g., the one I just described here) seems to be his research area so that’s naturally what he’d gravitate to.

We did not discuss measure theory in this class. I think the only measure theory courses we have at Berkeley are STAT 205A and 205B? I surprisingly couldn’t find any others by searching the math course catalog (is there really no other course on measure theory here???). But even Professor Aldous’ 205A website says:

Sketch of pure measure theory (not responsible for proofs)

Well, then, how do statistics PhD students learn measure theory if they’re “not responsible for proofs”? I assume they self-study? Or is measure theory needed at all? Keener says it is needed to properly state the results, but it has the unfortunate side effect of hindering understanding. And Andrew Gelman, a prominent statistician, says he never (properly) learned it.

As should be obvious, I do not have measure theory background. I know that if \(X \sim {\rm Unif}[0,1]\) then the event that \(X=c\) for any constant \(c\in [0,1]\) has measure 0, but that’s about it. In some sense this is really frustrating. I have done a lot of differentiating and integrating, but before taking this class, I had no idea there were differences between Lebesuge and Riemannian integration. (I took real analysis and complex analysis in undergrad, but I can’t remember talking about Lebesuge integration.)

Even in a graduate-level class like this one, we made lots of simplifying assumptions. This was the header text that appeared at the top of our problem sets:

For this assignment (and generally in this class) you may disregard measure-theoretic niceties about conditioning on measure-zero sets. Of course, you can feel free to write up the problem rigorously if you want, and I am happy to field questions or chat about how we could make the treatment fully rigorous (Piazza is a great forum for that kind of question as long as you can pose it without spoilers!)

Well, I shouldn’t say I was disappointed, since I was more relieved if anything. If I had to formalize everything with measure theory, I think I would get bogged down with trying to understand measure this, measure that, the integral is with respect to X measure, etc.

Even with the “measure-theoretically naive” simplification, I had to spend lots of time working on the problem sets. In general, I could usually get about 50-80 percent of a problem set done by myself, but for the remainder of it, I needed to consult with other students, the GSI, or the professor. Since problem sets were usually released Thursday afternoons and due the following Thursday, I made sure to set aside either Saturday or Sunday as a day where I worked solely on the problem set, which allowed me to get huge chunks of it done and LaTeX-ed up right away.

The bad news is that it wasn’t easy to find student collaborators. I had collaborators for some (but not all of) the problem sets, and we stopped meeting later in the semester. Man, I wish I were more popular. I also attended a few office hours, and it could sometimes be hard to understand the professor or the GSI if there were a lot of other students there. The GSI office hours were a major problem, since another class had office hours in the same room. Think of overlapping voices from two different subjects. Yeah, it’s not the best setting for the acoustics and I could not understand or follow what people were discussing apart from scraps of information I could get from the chalkboard. The good news is that the GSI, apart from obviously knowing the material very well, was generous with partial credit and created solutions to all the problem sets, which were really helpful to me when I prepared for the three-hour final exam.

The day of the final was pretty hectic for me. I had to gave a talk at SIMPAR 2016 in San Francisco that morning, then almost immediately board BART to get back to Berkeley by 1:00PM to take the exam. (The exam was originally scheduled at 8:00AM1 (!!) but Professor Fithian kindly allowed me to take it at 1:00PM.) I did not have any time to change clothes, so for the first time in my life, I took an exam wearing business attire.

Business attire or not, the exam was tough. Professor Fithian made it four broad questions, each with multiple parts in it. I tried doing all four, and probably did half of all four correct, roughly. We were allowed a one-sided, one-page cheat sheet, and I crammed mine in with A LOT of information, but I almost never used it during the exam. This is typical, by the way. Most of the time, simply creating the cheat sheet actually serves as my studying. As I was also preoccupied with research, I only spent a day and a half studying for the final.

After I had turned in the exam, Professor Fithian mentioned that he thought it was a “hard exam”. This made me feel better, and it also made me wonder, how well do professors generally do on their own exams? Well, assuming the exam was the same difficulty and that they didn’t directly know the questions in advance.

It was definitely a relief to finish the final exam, and I rewarded myself with some Cheeseboard Pizza (first pizza in ages!) along with a beautiful 5.8-mile run the following morning.

Now that I can sit back and reflect on the course, STAT 210A was enormously time-consuming but I learned a lot from it. I’m not sure whether I would have learned more had I had more of a statistics background; it might have made it easier to go in-depth into some of the technically challenging parts. I probably could have spent additional focused time to understand the more confusing material, but that applies to any subject. There’s always a tradeoff: should I spend more time understanding every aspect of the t-test derivation, or should I learn how to not be an incompetently bad C programmer, or should I simply waste time and blog?

One of the challenges with a technical course like this one is, as usual, following the lectures in real time. Like most of my Berkeley classes, I used sign language interpreting accommodations. No, they were not happy with interpreting this material, and it was hard for them to convey information to me. On the positive side, I had a consistent group so they were able to get used to some of the terminology.

In general, if I did not already deeply understand the material, it was very difficult for me to follow a lecture from start to finish. I can understand math-based lectures if I can follow the math derivation on the board, but with many of the proofs we wrote, we had to skip a lot of details, making it hard to get intuition. There are also a lot of technical details to take care of for many derivations (e.g., regarding measure theory). I wonder if the ability to fill in gaps between math and avoid getting bogged down with technicalities is what separates people like Professor Fithian versus the average statistics PhD student who will never have the ability to become a Berkeley statistics faculty member.

Regarding the quality of instruction, some might be concerned that since Professor Fithian is a new faculty member, the teaching would be sub-par. For me, this is not usually much of an issue. Due to limitations in sign language and my ability to hear, I always get relatively less benefit out of lectures as compared to other students, so direct lecture-teaching skill matters less to me as to whether there are written sources or sufficient office hours to utilize. (To be fair, it’s also worth noting that Professor Fithian won several teaching awards at Stanford, where he obtained his PhD.)

In case anyone is curious, I thought Professor Fithian did a fine job teaching, and he’s skilled enough such that the teaching quality shouldn’t be a negative factor for future students thinking of taking 210A (assuming he’s teaching it again). He moves at a relatively moderate pace in lectures with standard chalkboard to present details. He doesn’t use slides, but we occasionally got to see some of his R code on his computer, as well as his newborn child on Facebook!

I would be remiss if I didn’t make any suggestions. So here they are, ordered roughly from most important to least important:

  • If there’s any one suggestion I can make, it would be to increase the size of the chalkboard text/notes. For optimal positioning of sign language interpreting accommodations, I always sat in the front left corner of the room, and even then I often could not read what was on the far right chalkboard of the room. I can’t imagine how much harder it would be for the people in the back rows. In part, this may have been due to the limitations in the room, which doesn’t have elevated platforms for the back rows. Hopefully next semester they figure out better rooms for statistics courses.

  • I think the pace of the class was actually too slow during the first part of the course, about exponential families, sufficiency, Bayesian estimation, and so on, but then it became slightly too fast during the details on hypothesis testing. My guess is that most students will be familiar with the details on the first set of topics, so it’s probably fair to speed through them to get to the hypothesis testing portion, which is probably the more challenging aspect of the course. Towards the end of the semester I think the pace was acceptable, which means it’s slightly too fast for me but appropriate for other students.

  • Finally, I hope that in future iterations of the course, the problem sets are provided on a consistent basis and largely free of typos. We went off our once-a-week schedule several times, but still found a way to squeeze out 11 problem sets. Unfortunately, some of the questions either had typos or worse, major conceptual errors. This often happens when Professor Fithian has to create his own questions, which is partly out of necessity to avoid repeating old questions and risk students looking up solutions online. I did not look up solutions to the problem set questions, though I used the web for the purposes of background material and to learn supporting concepts (Wikipedia is great for this, so are other class websites which use slides) for every problem set. I’m not sure if this is standard among students, but I think most would need to do so to do very well on the problem set.

Well, that’s all I want to say now. This is definitely the longest I’ve written about any class at Berkeley. Hopefully this article provides the best and most comprehensive description of STAT 210A in the blogosphere. There’s a lot more I can talk about but there’s also a limit to how long I’m willing to write.

Anyway, great class!

  1. Professor Fithian: “sorry, not my idea.” ↩