[EM] Spatial models -- Polytopes vs Sampling

[Kristofer Munsterhjelm]

On 01.02.2022 04:35, Daniel Carrera wrote:
> 
> So I started reading up on quasi-MC. I haven't done this before so my
> initial thoughts might be incorrect, but I think I've found two things:
> 
> 1) It looks easy to do. There are libraries (e.g. scipy.stats.qmc) that
> just give you a sequence of numbers that you can use as a drop-in
> replacement for your random number generator in your MC integrator.
> 
> 2) But I see a lot of warnings saying that QMC constructions are
> designed for special values of n, like powers of 2 or large primes, and
> if I sub-sample or pick the wrong number of points I might ruin all
> their properties. This is potentially a huge issue because integrating
> over the polytope consists of throwing away all the points that aren't
> inside the polytope.

I'm hardly an expert on these matters, but I found a paper suggesting a
way to transform a simplex into a hypercube (by, I think, using extended
barycentric coordinates as a change of bases). If that's a workable
approach, then you could decompose the polytope into simplices,
transform these simplices into cubes, and do QMC on the cubes without
throwing any points out.

https://istam.iitkgp.ac.in/resources/2014/proceedings/59-istam-nm-fp-268.pdf

> Presumably that means that the Halton sequence isn't very finicky about
> exactly which points get included in the integral. As luck would have
> it, the Halton sequence is apparently superior for "low-dimensional"
> problems N <= 6 (Morokoff & Caflisch 1995), which is the dimensionality
> I most care about anyway. Honestly, the N=2 and N=3 problem is the most
> interesting for real world elections.

For 2D, you could compare your approach with
https://math.stackexchange.com/a/2283927 and the "exact approximate"
approach suggested by Thomas Ahle in a comment to
https://math.stackexchange.com/questions/627720; and see which is most
accurate/faster, if you have the time.

-km