Vector calculus in statistics

Question

I'm teaching a class on integration of functions of several variables and vector calculus this semester. The class is made up most of economics majors and engineering majors, with a smattering of math and physics folks as well. I taught this class last semester, and I found that a lot of the economics majors were rather bored during the second half. I was able to motivate multiple integrals by doing some calculations with jointly distributed random variables, but for the vector analysis part of the course the only motivation I could think of was based on physics.

So I'm wondering if anybody knows a statistical/probabilistic interpretation of any of the main theorems of vector calculus: Green's theorem, Stokes' theorem, and the divergence theorem. Part of the problem is that vector fields don't seem to come up very often in probability theory, let alone divergence, gradient, or curl. I also posted this question on math.stackexchange a few days ago, but I'm still looking for more ideas.

Answer 1

I doubt many statisticians will have to use vector calculus as it is taught for physics and engineering. But for what it's worth here are a few topics that would use it, at least tangentially. The underlying theme here is that holomorphic functions from complex analysis, which are composed of harmonic functions, are intimately linked through the Cauchy Riemann equations to both Stokes' and Green's theorems. These functions can be studied both by examining the the interior of their domain along with their boundary.

Probability Currents. This isn't just for quantum mechanics. In general, probability diffusions arise when studying time-varying probability distributions which change smoothly. This includes stochastic version of classical systems, such as the heat equation, Navier Stokes for fluid dynamics, wave equations for quantum mechanics, etc. Examples of equations include the Fokker-Planck equation and Kolmogorov Backwards/Forwards equations involve divergences, which in turn relate to heat equations, Feynan-Kac integrals, dirichlet problems and Green's functions. The keywords here are complex harmonic functions, which satisfy the mean value property, which in turn is a consequence of Green's integral theorem and Stokes' theorem. A classical example is calculating the exit time of a diffusion from a closed region, which reduces to evaluating integrals on the boundary of the surface and exploiting harmonicity within the region.

The main example here is problems involving Brownian motion, and in general the wide class of Ito Diffusions. A wonderful (and eccentric!) book on this is Green, Brown and Probability by the legendary Kai Chung.

The Disintegration Theorem for probability is implicity Stokes' Thoerem, in that one disintegrates a 3 dimensional probability measure onto the boundary of the surface that encloses its support.

In statistical mechanics and in markov random fields, there is a large prevalence of conservation in the form of currents. The Ising Model, especially at criticality, and its relatives can be studied from the point of view of discrete harmonic and holomorphic functions. From the Cauchy Riemann equations, one recovers both Green's Theorem and Stokes's theorem, in that currents are both divergence free and curl free, which together imply that the underlying field is holomorphic. A great reference on this is from the work of Smirnov, Chelkak and Dominil-Copin.

Answer 2

One example you could look into is quasi-likelihood. The discussion of these in McCullagh & Nelder: Generalized Linear Models uses (for the theoretical part) gradients and path-integrals in an essential way! See chapeter 9 of that book.