Group Theory AND Statistics


Diaconis does something cool, video at 11

Persi Diaconis is an interesting character. He dropped out of high school at 14 to become a professional magician, got a Ph.D. from Harvard, won a MacArthur “genius grant,” and is now a mathematician and statistician at Stanford University.

He also does interesting math; today I’ll talk about an example from a paper1 of his that I found particularly clever.

Counting voter rankings

Suppose we are holding an election between five candidates, and we ask voters to rank all five in order of preference. We’d like to take those rankings of the five candidates and turn it into a table where the (i, j) entry is the number of voters who ranked candidate i in place j.

Note that a ranking of five candidates can be thought of as a permutation of length five, an element of the group \( S_5 \). That is, the permutation \((5 4 3 2 1)\) corresponds to ranking candidate 5 first, candidate 4 second, etc. Next, we use the usual linear representation of \( S_5 \); for any given permutation \( \pi \), let \( \rho(\pi) \) be the 5 x 5 matrix with (i, j) entry equal to one if \( \pi(i) = j \), zero otherwise.

Define a function \( f: S_5 \rightarrow \mathbb{N} \) such that \( f(\pi) \) is the number times the vote \( \pi \) was cast. Hence, if 10 people all voted \((5 4 3 2 1)\), then \( f( (5 4 3 2 1) ) = 10 \). Now, take the Fourier Transform \[ \hat{f} = \sum_{\pi} f(\pi) \rho(\pi) \] Note that \( \hat{f} \) adds up the matrix for each vote times the number of times that vote appeared; this means that \( \hat{f} \) gives us a “table” (really a matrix) where the (i, j) entry is the number of times voters ranked candidate i in place j. This is exactly what we want!

Pithy Python program performs promptly

Here’s some Python code that computes this.

First off, some useful imports.

from collections import Counter
from itertools import permutations
import numpy as np

Next, we define our representation function \( \rho \).

def rho(p):
    return np.matrix([np.eye(len(p), dtype=int)[:, i] for i in p])

After that, some (wicked short!) code to compute the transform \( \hat{f} \).

def fft(ps):
    c = Counter(tuple(p) for p in ps)
    return sum(rho(p) * c[p] for p in c)

The following code also works, but it doesn’t technically follow the definition above since it hides \( f \).

def fft2(ps):
    return sum(map(rho, ps))

A quick function to normalize a table of counts to percentages instead:

def table(xss):
    return 100 * xss / np.sum(xss, axis=1)

And a quick check to see it in action:

if __name__ == '__main__':
    N = 5
    S_N = np.array(list(permutations(range(N))))
    ps = S_N[np.random.randint(np.factorial(N), size=500)]
    fhat = fft(ps)
            ps, fhat, table(fhat)))


I’m very excited about this; using group theory to better understand statistics and machine learning is exactly what I’ve been hoping to do since I finished grad school.

  1. “Applications of Group Representations to Statistical Problems,” 1990; available here. ↩︎