[EM] Spatial models -- Polytopes vs Sampling

[Kristofer Munsterhjelm]

On 02.02.2022 12:41, Daniel Carrera wrote:
> So I implemented my proposed "voter mass function" based on integrating
> "voter density" across a polytope. I have run some benchmarks on a
> sample 2D polytope (small dimension, but relevant to real elections). I
> implemented regular Monte Carlo (MC) integration, as well as the Quasi
> Monte Carlo (QMC) suggested by Kristofer. The QMC method was done with
> both a Halton and a Sobol sequence. The voter density function is a
> Gaussian distribution.

Great!

I was browsing the scipy.stats.qmc manual and noticed it has a third
method, Latin hypercube, explicitly designed for cubes. Would this
method be applicable to your problem, if you use the function g that's
zero outside of the simplex and Gaussian inside it? Does the proportion
of the cube occupied by the simplex vanish too quickly as d increases?

I would also imagine that you could reduce the dimension by one by using
a standard 1D Gaussian integral over the last dimension as long as you
can do line-simplex intersections to determine what line you should
integrate over. But perhaps the general covariance problem you mentioned
earlier would make this impractical - that it would be rather difficult
to line up the Gaussian integral with that line in the remaining dimension.

On a related note, I was reading James Green-Armytage's paper about
strategic voting: http://jamesgreenarmytage.com/strategy-utility.pdf. On
page 21, he states that an 8D spatial model is a good fit to the
political poll model, while 1D is not quite as good. He doesn't mention
intermediate dimensionality models, but it may provide a reason for
supporting high dimension spatial models (as long as the fit keeps
improving even when going from say, 7D to 8D). It does, I think, provide
pretty good evidence that there's little need for going beyond 8D, at least.

-km