[EM] Exact spatial model probabilities?

[Daniel Carrera]

On Tue, Jan 25, 2022 at 3:26 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> Perhaps the better approach is to use MC or quasi MC. Or meet in the
> middle somehow, e.g. let f(A>B>C, A.x, A.y, B.x, B.y, C.x, C.y) be the
> fraction of voters who vote A>B>C when the candidates are at the given
> coordinates; then use billiard sampling to determine f, and sample using
> MC or a perturbed over the coordinates. This could perhaps be more
> accurate than using MC for both picking the candidate locations and
> counting voter proportions.
>

I'm not familiar with billiard sampling. Is this it?

https://www.sciencedirect.com/science/article/pii/S037722171400280X

I have an idea that might be very naive, but here it goes. The problem is
hard in part because you want to compute f(A>B>C) for every permutation of
candidates. You could make a quicksort-like algorithm to iteratively
partition the space:

1) Start with a population of V voters and C candidates spread in some
N-dimensional space.

2) Select any two candidates {A,B}. Partition the voters based on whether
they are in the A>B camp or the A<B camp. Now you have two disjoint groups.
Partition each group based on where they fall on the {A,C} contest.
Partition those based on where they fall on the {B,C} debate, and so on.

The total number of operations is V*C^2 which... can be a lot if V is
large. But it might not be too bad if the alternative way of computing the
volume of 1 set of preferences costs more than doing a simple {A,B}
comparison V times. There might be some clever geometrical tricks to
quickly classify some of the voters, but I can't think of any right now
that is obviously faster.

Cheers,
--
Dr. Daniel Carrera
Postdoctoral Research Associate
Iowa State University
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