PhD Chances, math and stats

[Bourbaki1123]

For the key stuff, just glance at the bold statements and my program choices.

So; I tend to think that my stats are pretty iffy, but I'd like to see if I can get any input. I'm applying to a variety of programs (though I'm only putting up the math and stats choices here) because I've got a wide variety of interests out side of class (which accounts for my so-so grades, that and I really was not putting forth my best effort, but that's not really something I can change at this point). Some of my GPA could possibly be viewed as reasonable because of the heavy course load, advice on this would be much appreciated.

I go to a small, generally unknown university.

Here are my courses:

Calc 1 - B -Eh
Calc 2 C - Ouch, that semester was terrible overall. My GPA discounting my second semester is about 3.6, including it, it is about 3.4. I tutored calculus at my university for a year following this, so I suppose it makes up for it some.

Third Semester: I took 16 credit hours, all math

Calc 3 Eh, once again. I was pretty bad about doing homework.
Linear Algebra [A]
Intro to Higher Mathematics [A]
Abstract Algebra [A] (Yes, took this all at once, I got special permission for this course because I caught up to the work within three weeks and aced the first test)

Mathematics Software Programming [A]

Fourth Semester: 18 credit hours, 12 in math

Elementary Number Theory [A]
Abstract Algebra II [A]
Diff Eq
Discrete Mathematics [A]

Fifth Semester:
Grad level theory of Computation - [A], totally aced every test with top grade
Numerical Analysis [C] Ouch again, really slacked off.
Real Variables [A-]
Intro to algebraic geometry, Independent study [A]

Sixth Semester: Might have overdone it here, 20 credit hours of hard courses

Grad level Mathematical Logic I and II (quarter system at other uni) [B+] in both

Undergrad Complex Analysis [C+] Ouch
Advanced Calc 2
Applications of Algebra, independent study [A]
Object oriented Programming I [B+]
Ocaml and Bridge AI, Independent study [B+] Not enough programming experience on my end to pick up a functional language on my own, much less do original work in coding a Bridge AI. Also prof didn't know the language ahead of time. Probably an ill conceived endeavor all around...
And due to personal reasons (which I suppose I should explain in my statement of purpose) the only mathematics/computer science course I've taken in the last year is a Computational Complexity graduate seminar with the same prof who I did the mathematical logic courses with. I've chatted with him quite a bit, and actually think he may give me a good letter of recommendation, despite my so-so grades in his course, though I'm not sure.

This semester I'm hopefully going to be taking three grad courses; one in logic methods in computer science, one in graduate complex analysis (to make up for slacking off in the undergrad course) and one in graduate mathematical statistics. I'm three sigmas confident I could make A's in them if I actually do the work, and I plan to do it.

General GRE

660 V 790 Q 4.5 AW

Putnam: Scored a 10, taking again, hoping for a 20.

Math Subject: Will take in October, projected to be above the 80%ile, scored ~85th on the two re-normed practice tests; I figure if I put in a solid amount of studying over the next couple of months I should get at least that. Assume for the purposes of this assessment that it's 85%ile.

Research: University research fellow in Algebraic Cryptology one summer.

As far unquantifiables go, I've got a solid background in algebra and logic. I've got a lot of background in proof theory, which is a pet interest of mine. I've read a good bit of the Handbook of Proof Theory, and I've currently been reading up on Martin-Löf type theory because I'm interested in Voevodsky and Awodey's Homotopy type theoretic 'foundations' they've been setting up in Coq. I've got an interest in game semantics for proof theory as well, and I've been looking into Girard's Linear Logic as well as efforts in computability logic (such as cirquent calculus).

In addition to this, I've developed a strong interest in mathematical statistics and will be applying to a number of stats programs. My interests are in rational choice theory, Bayesian decision theory, machine learning, monte carlo methods, graphical models and experimental design. I've been working through E.T. Janye's Probability Theory, Bishop's Pattern Recognition and am getting Schervish's Theory of Probability.

Another side interest is in cognitive science and computational neuroscience. I've been reading quite about about that subject; parituclarly in The handbook of Brain Theory and Neural Networks, the MIT Encyclopedia of Cognitive Sciences, and Theoretical Neuroscience by Dayan and Abbott.

In my statement, I think that I could be quite specific about my research interests in Algebraic geometry, Logic and Neuroscience, a bit less so with statistics.

So bearing all of that in mind, could you guys give me an honest assessment for:

Mathematics:

CMU (specifically for the center for applied logic)
UIUC (for algebraic geometry)
University of Utah (alg geometry again)
University Notre Dame (for logic)
Rensselaer Polytechnic (applied math)

Still thinking about a few, I'll definitely apply to more than these.

Statistics:

Ohio state
Yale
University of Florida
University of Indiana
UCLA
Still researching schools for statistics.

Truthfully, if I can get into a stats program that looks really interesting I would probably choose that over a good math program.

Request: I really want to know whether I have a better shot in statistics than in mathematics, and whether my course choices are overly ambitious or not ambitious enough for either group above.

 

[Pyrrhus]

Other good stat schools are University of Minnesota, and Stanford University.

 

[Bourbaki1123]

Other good stat schools are University of Minnesota, and Stanford University.

Yes, those are fantastic stats schools, but would I have any chance at getting in with my profile? I assumed everything in the top 20 would be out of reach. I mean, truthfully my absolute dream school would be Carnegie Mellon for either stats or machine learning, but I didn't think I could get into either program, looking at grad student profiles and rankings.

 

[Pyrrhus]

A good method is to apply to at least 5 schools. 2 top 10, 2 top 20, and 1 safety. I actually did 8, where 3 were top 10, 3 top 20, and 2 safety. I am not sure about the profile, they're looking, but results vary.

I think your GPA is decent, and if you have significant research experience you should be better. Also, your GRE General is good, so it won't hurt you (GRE generals rarely add to the application, low scores just hurt it). I guess you should work hard for your GRE Subject, so you can make your application look better.

 

[Bourbaki1123]

So, I'm still wondering; are stats programs in general less competitive than mathematics programs? I get the impression that the profile for a lot of students at top stats school look quite a bit like profiles of students at math students at schools of lower rank. For instance, looking at Stanford, the average Math subject GRE is 82%ile, whereas for the math department I would assume it is somewhat higher, certainly for Berkeley's mathematics program they expect you to get at least an 80%ile.

Subjectively, statistics programs seem somewhat less intimidating than math programs, at least from the standpoint of a math major. Then again, maybe I'm looking at the wrong factors. I sort of assumed that math GRE would be even more relevant for stats/applied math because of the heavy weighting towards calc, linear algebra, probability and real analysis content.

 

[Pyrrhus]

Note that, I am actually in Economics at University of Minnesota, so I don't know about admissions explicitly in Stats. However, I did once pursue a Master in Stat here, but I later decided to drop it because it was too much requirements beyond just the theoretical sequence which I had completed. My experience from the classes is that the Stat department here is quite good.

 

[Stengah]

I've been looking at statistics grad schools this summer, so what I have to offer is purely based off of reading many grad program websites for a variety of schools. Also I have been focusing on biostats a bit more. I asked in a thread a few weeks ago about the competitiveness of grad programs in stats compared to physics or math, but no one responded. But I would guess that they are a bit less competitive, with the top schools getting around 150-200 applications for roughly 15-20 spots. Although I'm not sure what the numbers look like for math. It also seems that only a handful (the top schools) require the math subject GRE for admission consideration. I suspect that schools need a broader range of applicants and they aren't all expected to be that well prepared in pure math to take the subject test. I have seen a few that say you only need 3 semesters of calculus and 1 of linear algebra during your undergrad years. I'm sure the common accepted applicant has much more math than that, but you usually don't see analysis classes as a requirement prior to admission. I do agree that they seem less intimidating than pure math, but I'm in the same position as you, I don't really know for sure. I've seen some schools say that up to 60% of their stats students are internationals, and I don't know what the norm is for this, but that seems like a lot. As for whether or not you have a better chance in statistics, I'm not sure. You haven't taken many actual stats classes, but you have plenty of advanced math that will have you well prepared. It almost seems like getting an undergrad degree in statistics is kind of a joke, but many schools do offer them so you will face those applicants. But for example my school offers a masters in statistics, but no undergrad degree in statistics. You can only get a B.S. in math and then "emphasize" in statistics. Because of this I suspect that is better to take as much math as you can then take some statistics on the side. It also appears that statistics is really a smaller field than math, if you don't consider it a sub-field. There are those who don't consider it a sub-field. And for a biostats example, UW says they accept biology majors to their program. In conclusion, a heavy load of math doesn't seem to be the norm for admission to lower ranked schools, but it should look really good if you have it. In general statistics just doesn't seem like a very popular choice for people, I'm not sure why though. As for me, I'm considering applying to statistics graduate programs as well, depending on how much I get interested, but I'm a physics major.

 

[Bourbaki1123]

You know, I think that there are a few things that seem to contribute to the lack of popularity in statistics.

(1) It doesn't have the same mystique as pure math or theoretical physics (No Feynman, not John Nash, not as much of a romantic history, at least as far as I can tell, which makes sense given its relative age I guess)

(2)I think that a lot of people don't really look into the foundations and the philosophy of it, which is, IMO, a huge part of what makes it so attractive. E.T. Jaynes' book has really given me a lot of interest in the field, and the applications to machine learning and rational choice theory are extremely interesting to me. I really enjoy decision theoretic problems, and Bayesian decision theory is very attractive to me.

(3) I think there is a certain feeling (and I think this is more coming from the pure math perspective) of discomfort/negativity towards the field because undergraduate frequentist stats feels too arbitrary while at the same time lacking real mathematical rigor, i.e. not being quite so proofy. Doing stats doesn't quite feel like doing other math, and at the same time it doesn't feel quite as natural as say, applying calculus to physics problems.

For me the notion of machine learning, and that learning is extracting patterns from information and making hypotheses about those predictions in order to maximize utility, really was what turned me on to statistics. That is why statistical decision theory and statistical learning theory are my biggest interests. There is just so much philosophical appeal to the field when you start viewing things in that way. You see the breadth of deep applications and the beauty of the theory.

At least, that is my take.

 

[Stengah]

Yes I agree with everything you said. Probability theory is what originally got me interested in the field, but nearly everything else I read sounds interesting as well. I'll be taking my first statistics course this fall though so I'll see where it takes me. One other thing I forgot to mention is statistics in a high school setting. At my school, if you took statistics, you were seen as not good enough for calculus. Taking stats was kind of the "easy way out" into taking more math. I think this gives it a negative appeal early on for many people. I never took it, but I heard it was very boring. I'm guessing in high school it would just be too early to introduce any interesting or complicated concepts. Or maybe you need to reach a certain maturity level to really appreciate it.

 

[Pyrrhus]

You know, I think that there are a few things that seem to contribute to the lack of popularity in statistics.

(1) It doesn't have the same mystique as pure math or theoretical physics (No Feynman, not John Nash, not as much of a romantic history, at least as far as I can tell, which makes sense given its relative age I guess)

What?! What about Fisher? Pearson? or even Bayes!. Come on!

and I can't believe I forgot Kolmogorov!.

A favorite to me is Harald Cramer and how it joined the russian school of theoretical statistics with the british school of practical statistics.

 

[Bourbaki1123]

What?! What about Fisher? Pearson? or even Bayes!. Come on!

and I can't believe I forgot Kolmogorov!.

A favorite to me is Harald Cramer and how it joined the russian school of theoretical statistics with the british school of practical statistics.

So mathematicians get to keep Ramsey right? ;)

 

[Pyrrhus]

So mathematicians get to keep Ramsey right? ;)

Hehe, no. We get to keep them all even Debreu

 

[chiro]

Hey Bourbaki.

I read your courses and I can't see any solid major courses like inference, applied probability, experimental design, linear models (GLM's would be good) or any other elective courses.

I don't about the US, but in Australia to do the decent Masters (and subsequently PhD courses) you need a major in statistics with decent marks in your later years. You can get into PhD courses here with a good honors degree, but an honors year will have most of the masters subjects anyway.

There are Masters courses for people without the major subjects in statistics but that takes longer because you have to do some of the pre-requisites that most of the stats majors have done.

I don't know if people think that statistics is not as hard as say pure math or physics, but regardless of that answer you need to realize that statistics isn't just probability mean and standard deviation: it is a deep subject and requires a lot of experience (and coursework) to become proficient in it.

I'm not saying you can't do it, but I would be surprised if you got into a PhD stats program without at least the basics mentioned above.

 

[twofish-quant]

I think there is a certain feeling (and I think this is more coming from the pure math perspective) of discomfort/negativity towards the field because undergraduate frequentist stats feels too arbitrary while at the same time lacking real mathematical rigor, i.e. not being quite so proofy.

I think the reason that statistics is taught this way is that there is a huge number of jobs in which you need someone to go in and just calculate stuff. So a lot of undergraduate statistics courses end up being "cookbook exercises" where you are taught to do X, Y, and Z by the book so that you can do the same thing once you get your bachelors and go to work shoveling numbers for a biotech company.

Doing stats doesn't quite feel like doing other math, and at the same time it doesn't feel quite as natural as say, applying calculus to physics problems.

I think this is largely an artifact of how statistics/calculus/physics is taught and how it is used in industry.

There is just so much philosophical appeal to the field when you start viewing things in that way. You see the breadth of deep applications and the beauty of the theory.

You can get the same thing if you start thinking deeply about pretty common things. For example, when someone says that there is a 30% chance that they will get into Harvard, what does that exactly *mean*?

 

[Stengah]

Hey Bourbaki.

I read your courses and I can't see any solid major courses like inference, applied probability, experimental design, linear models (GLM's would be good) or any other elective courses.

I don't about the US, but in Australia to do the decent Masters (and subsequently PhD courses) you need a major in statistics with decent marks in your later years. You can get into PhD courses here with a good honors degree, but an honors year will have most of the masters subjects anyway.

There are Masters courses for people without the major subjects in statistics but that takes longer because you have to do some of the pre-requisites that most of the stats majors have done.

I don't know if people think that statistics is not as hard as say pure math or physics, but regardless of that answer you need to realize that statistics isn't just probability mean and standard deviation: it is a deep subject and requires a lot of experience (and coursework) to become proficient in it.

I'm not saying you can't do it, but I would be surprised if you got into a PhD stats program without at least the basics mentioned above.

I think the US is just different than Australia. Statistics majors aren't as common here so grad schools often accept people coming from pure math or other sciences. Some schools don't even require any statistics background as long as you have done the calculus sequence and linear algebra, and maybe some advanced analysis.

 

[Bourbaki1123]

I don't know if people think that statistics is not as hard as say pure math or physics, but regardless of that answer you need to realize that statistics isn't just probability mean and standard deviation: it is a deep subject and requires a lot of experience (and coursework) to become proficient in it.

I listed my research interests and books I've been reading; I don't know where you get the idea that I might think that stats is just about sample mean and standard deviation. I'm familiar with the fundamentals of measure theoretic probability theory, basic sampling theory and hypothesis testing; I'm aware of both Bayesian (conjugate priors, Beta/gamma distributions, Dirichlet distribution, Shannon entropy ) and frequentist methods in statistical inference, and I'm familiar with HMMs, Monte Carlo methods and Bayesian networks as well (I've been reading Judea Pearl's Causaility, and Probabilistic Graphical Models: Principles and Techniques).

I'm also familiar with Solomonoff induction, Kolmogorov complexity and statistical complexity, which arguably reside more in the realm of probability theory, but at least one application has been to a general mathematical theory a a http://www.hutter1.net/ai/aixigentle.pdf" ).

In any event, if I though that statistics was all about checking the confidence interval for some goofy hypothesis about running out of widgets over a sampling distribution (like many undergraduate introductory courses seem to teach), I wouldn't be interested. I would go try to do my PhD in algebraic geometry or mathematical logic, or maybe in some sub-discipline of complexity theory. Not that I don't plan on applying to schools in those areas as well, I do; I just think that statistics has the potential to be more interesting and and that my life might be more enriched by the sorts of applications (such as working on machine learning, neural codes and genomics) that statisticians get to do.

ETA: Another significant factor here is that my university didn't offer a full B.Sc. in stats until about a year ago, so there wasn't/isn't exactly a good breadth of courses there. Also; I'm planning on taking a graduate mathematical statistics course as well as two under graduate stats courses (one in experimental design, one basic probability/stats course for math majors for my degree requirement) this semester at a neighboring uni, and that should be on my transcript when I'm applying

 

[Bourbaki1123]

You can get the same thing if you start thinking deeply about pretty common things. For example, when someone says that there is a 30% chance that they will get into Harvard, what does that exactly *mean*?

Right; or thinking of a plan for completing a project in terms of the product of the probability weights for each step working out correctly in order to properly weight the likelihood of project completion within a set time limit (http://en.wikipedia.org/wiki/Planning_fallacy" ). The intuitive notion that adding steps to a plan, or adding complexity to a hypothesis (thinking ockham's razor here) drop likelihood by some factor less than one is something that people don't seem to take into consideration in their everyday lives.

Even more clever, you can http://lesswrong.com/r/discussion/lw/71b/individual_deniability_statistical_honesty/" in order to get more accurate data, from which you can then infer some useful statistic.

 

[twofish-quant]

Even more clever, you can http://lesswrong.com/r/discussion/lw/71b/individual_deniability_statistical_honesty/" in order to get more accurate data, from which you can then infer some useful statistic.

Or you hire statisticians to do "anti-gaming" (google for it). What that involves is to make your actions look random so that other companies can't figure out what you are doing.

Also...

What happens in finance is that measures of human belief are effectively stated in terms of probability language. I sign a contract with you in which you pay me $1 if US defaults. How much you are going to pay for that contract is a measure of your belief, and how much the market is willing to pay for that is a measure of market beliefs.

What will get you in serious trouble is that if you mistake "probability as a statement of human belief" (Bayesian) with "probability that an event may happen." (frequentist) I may have a strong subjective belief that the aliens will land or mortgages won't default, but I may be delusional. And the market may have such a belief, but the markets may be delusional.

 

[Bourbaki1123]

Duplicate.