[EM] Spatial models -- Polytopes vs Sampling

[Forest Simmons]

El jue., 3 de feb. de 2022 2:34 p. m., Daniel Carrera <dcarrera at gmail.com>
escribió:

>
> On Thu, Feb 3, 2022 at 2:15 PM Kristofer Munsterhjelm <
> km_elmet at t-online.de> wrote:
>
>> 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?
>
>
> I tested it, and it didn't do as well as Sobol. The Latin Hypercube was a
> little faster than the Halton sequence but a little less accurate. I think
> Sobol has a huge advantage because (according to Wikipedia) there is a fast
> implementation based on low-level bitwise operations and that's just hard
> to beat with regular arithmetic.
>
>
>
>> Does the proportion
>> of the cube occupied by the simplex vanish too quickly as d increases?
>>
>
> I ran a test. It's certainly an issue, but not quite as bad as I had
> expected. First, the library can easily detect the polytopes that are
> completely empty. For the rest, I ran 10 sample elections with candidates
> uniformly random in the unit cube. The proportion of the cube occupied by
> the polytope seems to approach 10% as I get closer to 8 dims.
>
> N_candidates = N_dim + 1
>
> 2 Dims: v = 0.535417 +/- 0.236171
> 3 Dims: v = 0.256944 +/- 0.134119
> 4 Dims: v = 0.156250 +/- 0.065763
> 5 Dims: v = 0.134810 +/- 0.035892
> 6 Dims: v = 0.125693 +/- 0.009282
>
> That said, higher dimensions do get more expensive for other reasons ---
> like the fact that you have (N_dim+1)! polytopes. I was planning to run the
> simulation till N_dim=8 but I've been waiting for N_dim=7 to finish and it
> just doesn't want to finish.
>
>
> 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.
>>
>
> Yeah. If we assume that the Gaussian is fully symmetric (which feels a bit
> restrictive) you could imagine drawing radial lines and figuring out where
> they cross the polytope. The details could be a bit complicated. I have no
> idea if that would be faster, but I could try something like that. I can't
> work on this idea right now, but I didn't want to dismiss it.
>
>
> 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.
>>
>
If a poll has potential voters fill out a questionnaire with where they
self locate on 40 issue, you end up with a forty by n matrix M, where n is
the number of voters that submitted their completed questionnaires.

If you then diagonalize the 40 by 40 matrix that is the matrix product of M
by its transpose ... the eigenvalues along the diagonal with reveal the
relative significance of the respective issues. When you get to an
eigenvalue that is a order of magnitude smaller than the largest one, you
can conclude that the remaining issues were not as relevant to the voters
or else weaker clones (correlates) of the more
interesting/important/controversial issues.

Evidently James Green-Armytage made a judgment on where the drop in
"singular value" magnitudes became significant.

>
> It's good to have a maximum. I hope we can get away with fewer than 8
> dimensions. Regardless of the true political dimensions of the electorate,
> if you only have N<8 parties, those parties will lie in an
> (N-1)-dimensional subspace. That's why I suspect that we can get away with
> modelling a lot less than 8D and just be aware that I'm only modelling the
> subspace of political positions that is actually spanned by the candidates.
> The US is a specially pathological example, where apparently your views on
> LGBT rights somehow dictate your views on sex education, AR-15s, tax law,
> climate change, vaccines, and the Israel-Palestine conflict.
>

PR is supposed to help fix this, but ingrained traditions are hard to buck
... and (obviously) PR has no chance at all of helping if we never adopt it.

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