Importance of Measure Theory

[cyprusprior]

Having gotten into several different biostats PhD programs, I'm now trying to decide where to go. One of the main factors I find myself contemplating is the importance of measure theory as a "required" part of the PhD curriculum. From what I can tell, there is an active debate going on in many biostats, and even pure stats, departments about whether measure theory should be required (ie, included on the qualifying exams) or whether it should be an elective. I can see good arguments on either side. For example:

http://andrewgelman.com/2008/01/14/what_to_learn_i/

http://simplystatistics.org/2012/08/06/in-which-brian-debates-abstraction-with-t-bone/

http://simplystatistics.org/2012/08/08/on-the-relative-importance-of-mathematical-abstraction/

 

Even if I end up attending a PhD program where measure theory is not required, I do actually plan to take it just from my own curiosity. My main question is, does the presence/absence of measure theory in one's PhD curriculum affect academic job placement upon graduation? I can see a scenario where a department is considered less "rigorous" because they don't require measure theory. On the other hand, if the department produces students with, say, a stronger computational skill, maybe it would compensate? In a similar vein, can anyone give some concrete examples of how learning measure theory helped them in their later research, other than "developing abstract thinking skills"?

[StatPhD2014]

Having gotten into several different biostats PhD programs, I'm now trying to decide where to go. One of the main factors I find myself contemplating is the importance of measure theory as a "required" part of the PhD curriculum. From what I can tell, there is an active debate going on in many biostats, and even pure stats, departments about whether measure theory should be required (ie, included on the qualifying exams) or whether it should be an elective. I can see good arguments on either side. For example:

http://andrewgelman.com/2008/01/14/what_to_learn_i/

http://simplystatistics.org/2012/08/06/in-which-brian-debates-abstraction-with-t-bone/

http://simplystatistics.org/2012/08/08/on-the-relative-importance-of-mathematical-abstraction/

 

Even if I end up attending a PhD program where measure theory is not required, I do actually plan to take it just from my own curiosity. My main question is, does the presence/absence of measure theory in one's PhD curriculum affect academic job placement upon graduation? I can see a scenario where a department is considered less "rigorous" because they don't require measure theory. On the other hand, if the department produces students with, say, a stronger computational skill, maybe it would compensate? In a similar vein, can anyone give some concrete examples of how learning measure theory helped them in their later research, other than "developing abstract thinking skills"?

No one really cares about your transcript when you apply for academic positions

[Stat Assistant Professor]

Agreed with StatPhD2014. They won't look at your transcript. Hiring committees will look at your research statement, teaching statement, teaching portfolio, and cv, and make decisions based on these, plus your interview and your presentation of research (if you are invited to interview). At the place where I did my Masters, invited job candidates gave presentations that everyone in the department was invited to (grad students, faculty, etc.). Not sure to what extent the graduate students' feedback was weighed.

 

From my understanding, measure theory is most crucial for those who study probability theory. Applied statistical research areas probably have little use for it. I cannot imagine any statisitcal machine learning professors even remembering much from measure theory. The coursework in a PhD program is mostly to help you develop enough basic tools so you can pass qualifying exams (most material which will be forgotten later) and so you can gain a better idea of what research you want to do. My friends in pure math who are on the job market don't remember much of anything from their classes except for those related to their research area (so for instance, my friends who are algebraists or algebraic geometrists remember very little from analysis/measure theory even though they were required to take analysis in order to graduate and pass quals).

Edited February 18, 2014 by Stat Applicant

[mittensmitten895]

No one really cares about your transcript when you apply for academic positions

 

Even if hiring committees don't care about coursework, your curriculum can affect academic job opportunities by shaping your selection of research area, your approach to a chosen research problem, your ability to understand and incorporate relevant literature, etc.  I haven't taken measure theory yet, but people who think it is important seem to believe it is very important.  (BTW, cyprusprior, this is a great question.  I'm eager to hear what others have to say on the issue.)

[Stat Assistant Professor]

Is measure theory required for any biostats program?

 

It is required for many stats programs, but I see that some highly reputable programs that are more "applied" in nature don't even require it (e.g. Carnegie Mellon does not appear to require measure theory at all). I'm assuming some places lump probability theory with their mathematics depts (as CMU does), while keeping their statistics depts more applied in nature and separate from that.

 

I guess the importance of measure theory in the job search might depend on whether you apply for more theoretical stats or applied stats openings. I guess having measure theory would be helpful if you plan to apply for jobs in mathematics departments too, not just statistics depts.

[wine in coffee cups]

I think leaving a program knowing how to read an academic paper or text that uses measure theoretic statements of integrals and densities is important. But you don't really need a whole class to learn what is meant when something is written using dP or dmu notation. Would not say I gained anything statistically useful from going through tedious proofs of the Caratheodory extension theorem and all that. That said, I enjoyed the measure theory class taught in my program. Coming from a math background, it was more familiar territory for me than the statistical theory courses, and quite a bit easier.

 

My department has measure theory as a nominal requirement/prerequisite for the PhD theory courses, but I would not say it was necessary beyond passing familiarity with the notation, understanding that you can often ignore issues arising only on sets of measure zero, that expectation is integration with respect to a probability measure (which can be a weird mixture of continuous and discrete densities), and knowing when to invoke the big theorems and inequalities (Fatou's lemma, monotone convergence, dominated convergence, Fubini/Tonelli, Cauchy-Schwarz, Markov's inequality, etc.). I think most departments would do well to condense these ideas into a short crash course rather than use a whole quarter/semester actually proving all the results, and I suspect that ones that don't require measure theory already do exactly that as part of a required probability or theory sequence.

 

As to how this relates to choosing a program, I don't think you need to worry about the requirement/lack thereof affecting your ability to get a job. There are so many more important considerations beyond whether everyone else is required to take a math class you are planning to take anyway that this really should be at the bottom of your priorities. For example, something you maybe should care about is not so much specific requirements like one measure theory course, but how burdensome the collection of them are and the degree to which this interferes with becoming a researcher. My program used to have heavy coursework and qualifying exam requirements, which prevented students from getting involved in research until their third (or even fourth) years. This changed right before I started, so the more recent cohorts have more students involved in research during their first and second years, which hopefully will mean shorter times to degree for many of us and stronger CVs when applying for jobs.

[mittensmitten895]

Is measure theory required for any biostats program?

 

 ... kind of?  It's not as in depth as you would find in math or stat departments, but it is touched upon.  Michigan biostatistics does Casella and Berger for its first year MS theory courses, and students planning on a PhD currently take a 2-course sequence in the statistics department that covers some measure theory.  (I say "currently" because I've heard they are revisiting this requirement.)  The material from the higher level courses shows up on the qualifying exams.  Because I was not planning on a PhD in biostatistics until just a few months ago, I have not taken those classes.  So I know measure theory matters at least a little in biostatistics because it's delaying my quals by a year =).  I would guess you might also find measure theory in required courses at UNC biostatistics because of its theoretical orientation.

[adagietto]

Hopkins at least requires measure theory for all PhD students.

[Stat Assistant Professor]

Well, I guess it never hurts to have more theoretical understanding of the math and the ability to read articles that use sigma-algebras and Lebesgue integration. So it's not a bad idea to take measure theory.

 

But I don't think hiring committees will care at all if your research never touches upon it at all. They'll care about your research, and many topics in statistics do not use measure theory at all. I'm not sure that will be held against someone on the job market (unless the hiring committee is specifically looking for someone whose research is in the area of probability theory).