Measure Theory vs Bayesian Stats

[m0fazi0]

Hi everyone. I’m in a little bit of a pickle in choosing some courses for this year. 

A little background: I’m finishing up a MS Stat this year, and was hoping to apply for the Fall 2019 Stat PhD cycle. I have the opportunity to take a grad level Real Analysis/Measure Theory sequence, or a Bayesian Statistics sequence, but cannot do both.

My question is, with some mathematical background already (I completed a year of analysis in undergrad and did fine), would it be a wise choice to opt for the Bayesian courses? I haven’t yet taken a Bayesian course and have only scratched the surface in a theory course which mainly focused on the frequentist perspective. From what I’ve read, it seems like something that I might really enjoy doing more of, especially in a doctoral program (but I can’t really know unless I do it). However, I understand that perhaps my application would be stronger with measure theory under my belt before pursuing a PhD. This being said, the grade likely wouldn’t be posted until after applications are due, but only slightly.

If anyone has insight into my predicament or a similar experience, please don’t hesitate to chime in. Thanks.

[theduckster]

I am under the impression that most rigorous Stat PhD programs have their students take a sequence in graduate probability theory that heavily utilizes measure theory. In light of this, taking a grad level class in measure theory would have you fully prepared for the rigors of such curricula and maybe even start looking into research topics in pure probability right out the bat (if this is what you are interested in).

On the other hand, you mentioned that the programs probably will not see your grade (if they did, I would recommend measure theory hands down). Given this and your passion for Bayesian statistics (which is really important for industry and makes for fascinating research in its own right), you really can't go wrong either way.

[Stat Assistant Professor]

If you think you want to conduct research more on the probability theory side rather than statistics, then take the graduate analysis sequence. But if you are genuinely interested in Bayesian statistics and you think you might want to conduct research on it, I think that this sequence would be much more worth your while. Assuming your application is already very strong (e.g. other math classes, including undergraduate real analysis), I don't think it's really necessary to have a very deep knowledge of measure theory/abstract integration prior to enrolling in a Statistics PhD program. They usually teach you the measure theory they want you to know, and it's usually not even necessary to take a whole stand-alone class on it (unless your specialization is in probability theory rather than stat).

[bayessays]

I completely agree with the two people above. The only way I would lean towards the measure theory more is if you plan on doing extremely theoretical work (in which case you probably wouldn't be asking this) or applying to extremely theoretical schools (Chicago, Penn, etc).  I don't think otherwise it will be a big enough boost to your application to make up for the fact that it seems like the Bayesian class will be more of a boost to your development and more interesting to you. 

[Stat Assistant Professor]

It's worth noting that most students in Statistics PhD departments don't even take a full semester or year of stand-alone measure theory/abstract integration/functional analysis, UNLESS their PhD advisor recommends it (or unless they are just doing it for their own personal edification). In the department where I got my PhD from, the professors who specialized in Markov chain Monte Carlo theory *might* recommend their PhD advisees take functional analysis (since they need to work with Hilbert spaces and operator theory), and I'm sure the profs who do theoretical research on stochastic processes would also recommend this to their students. But just about every other PhD student I know only learned measure theory in their PhD probability class.