Establishing a Probability PhD Program: Prerequisites for Research

[Shackleford]

If you were to establish a curriculum for a Probability PhD program, what courses would you include as a prerequisite to doing research?

So far, I have: analysis, complex analysis, functional analysis, stochastic analysis, measure theory, and partial differential equations.

This presupposes entry-level calculus, probability and statistics, and linear algebra.

If your potential research topics included Bayesian computation (MCMC) or inverse theory (geophysics), what additional mathematics courses would you include?

 

[PAllen]

What is the probability of this program being offered?

 

[Vanadium 50]

What is the probability of this program being offered?

What is the Baysean prior of this program being offered?

 

[Shackleford]

What is the Baysean prior of this program being offered?

Good question. I suppose there should be a course on Bayesian statistics/inference.

 

[Shackleford]

What is the probability of this program being offered?

This program? Zero. I've only seen a PhD Probability program or two in the UK, which are generally research based without all of the prerequisite courses and myriad examinations.

 

[member 587159]

In general, rigorous probability theory is measure theory heavy. Therefore, a good knowledge of measure theory (and more generally analysis) is what you will need the most.

 

[Klystron]

Probably not all but some Probability candidates should emphasize electronics and the underlying physics not only of measurement but also software/hardware implementation such as random number generators, signal correction and automatic error correction. Or the math fields underlying such in keeping with the original question.

Perhaps I am suggesting an 'applied' as well as math-theoretic Probability doctorate where the former also designs measuring methodology?

 

[Shackleford]

In general, rigorous probability theory is measure theory heavy. Therefore, a good knowledge of measure theory (and more generally analysis) is what you will need the most.

That's my observation as well. To that end, I have A First Look at Rigorous Probability Theory by Jeff Rosenthal.

 

[Shackleford]

Probably not all but some Probability candidates should emphasize electronics and the underlying physics not only of measurement but also software/hardware implementation such as random number generators, signal correction and automatic error correction. Or the math fields underlying such in keeping with the original question.

Perhaps I am suggesting an 'applied' as well as math-theoretic Probability doctorate where the former also designs measuring methodology?

That's a good suggestion. I should add Numerical Analysis and maybe even a Signals course to the second question's answer. One could apply MCMC Bayesian inference to the error measurement methodology (or error function).

 

[StatGuy2000]

To the OP:

On a serious note, probability is a vast field within mathematics which can intersect a number of different research areas, from within mathematics (e.g. PDEs, harmonic analysis, topology, etc.), to physics (e.g. statistical mechanics), computer science (e.g. randomized algorithms, machine learning), and statistics. So what prerequisites are required to pursue research in probability will likely depend on what specific research field within probability the student will care to pursue.

There are certain basic requirements that all mathematics graduates intent on pursuing a PhD should learn, some of which you have already listed -- entry level probability and statistics, real analysis, complex analysis, functional analysis, measure theory, differential equations (both ordinary and partial), combinatorics (optional but recommended), linear algebra, algebra, topology, set theory (optional but recommended).

In terms of MCMC specifically, the question would be whether the focus will be on the theoretical properties of Markov chains themselves (in which case some background in statistical physics may help), theoretical properties of convergence rates of the algorithms themselves (in which case a background in theoretical computer science, notably theory of computation may help), or on application of said algorithms to Bayesian statistics (in which case advanced courses in statistics, specifically courses in Bayesian statistics, will help).

As for inverse problems, from my (limited) understanding that would fall under either non-parametric statistics rather than probability theory per se, or under harmonic analysis (I'm thinking specifically of applications of wavelets on inverse problems found in imaging). But others who are more knowledgeable can speak to this.

 

[Dr. Courtney]

I don't tend to like courses as prerequisites for research in any field. For me, this unnecessarily delays research, when my view is that it should start as soon as possible and need not be delayed artificially to fulfill other requirements - completion of certain courses, passing PhD qualifying exams, whatever.

Of course, the scope of one's research may be limited by completed coursework, as may the ability to be the lead on a project. But, when possible, students should begin research early.