[EM] Exact spatial model probabilities?

[Forest Simmons]

This reminds me of the n-body problem. For n not too large, the
Parker-Sochaski method is the best numerical solution technique that I know
of so far.

But for many years they have been using statistical mechanics in the many
body problem ... from n=3 on up to n =infinity ... thermodynamics,
magneto-fluid dynamics, etc.

Those guys know how to do it.

When I did minuteman missile simulations back in the early1970's we had a
Spherical Harmonics gravitational model with hundreds of terms, which was
necessary to get a ballistic missile to within a few feet of an enemy
missile silo half way around the world.

The guys that came up with it used data from satellite tracking
observations to get a spherical harmonics mass density model of the earth.
Once they had that, it was a simple matter to express the gravitational
potential in spherical coordinates, and from that, gravitational force via
the gradient of the potential.

We need spherical harmonics expansions of the voter density function on the
surface of a sphere of the right dimension.

Now, I know why the suggestion of a spherical distribution ... you can have
a uniform probability density on a compact manifold like a sphere or a
cartesian product of circles, but not for an open manifold like an entire
plane .. for that, the best (closest to smooth & uniform) you can do is a
Gaussian.

[Somebody should start doing sphericalYee diagrams.]

We need to get one of those geodesical scientists from the 1970's who
solved this problem without super computers ... they could tell us how to
do it ... just like the astronomers of the nineteenth century could use the
"method of orbital elements", a clever perturbation technique to do all of
the numerical integration of all the high precision n-body celestial
mechanics problems of the solar system (planets, moons, comets  asteroids,
etc) ... with nothing beyond mechanical adding machines and tables of trig
and log functions.

El dom., 23 de ene. de 2022 12:26 p. m., Daniel Carrera <dcarrera at gmail.com>
escribió:

>
> On Fri, Jan 21, 2022 at 9:02 AM Kristofer Munsterhjelm <
> km_elmet at t-online.de> wrote:
>
>> It's easy to approximately find such probabilities for spatial models as
>> well by just creating an instance and randomly placing voters on it to
>> determine what they vote, and hence in the limit of number of trials
>> going to infinity, get the probability.
>>
>> But I was wondering, is it possible to do so exactly? Let's say that in
>> a spatial model, first c candidates are placed uniformly at random in an
>> unit d-cube. Then voters are also placed uniformly at random in this
>> cube and they prefer candidates closer to them to candidates further away.
>>
>> Now I would suppose that the probability that a voter votes A first is
>> equal to the volume of the Voronoi cell that belongs to A. (And the
>> probability that a voter will vote A>B>C is the volume of the
>> intersection of the closest-neighbor Voronoi cell for A with the
>> second-closest-neighbor for B and the third-closest-neighbor for C.)
>>
>> Well, that eliminates one source of randomness -- assuming I can exactly
>> calculate the vertices of these regions. But there's still the second
>> source in the randomness of the candidates. Do you know of any calculus
>> tricks to get the probabilities over every possible candidate position,
>> or is this simply too hairy?
>>
>
>
> I have been thinking about this. I don't have a clever solution for you,
> but I have some thoughts that could be a stepping stone toward a solution.
> What you are trying to do is compute an integral in a highly dimensional
> space. Let's say that for a given set of candidates you have a function f()
> that outputs a real value. That might be a utility, or a probability, or
> whatever. I just want it to be a real number and not binary so that we can
> hope that f() is smooth in the space spanned by the candidates.
>
> The set of candidates can be thought of as a point in a C*D dimensional
> space where C is the number of candidates and D is the dimension of your
> D-cube. So you have a vector:
>
> x_jk = position of candidate j on issue k
>
> x = [ x11, x12, ... , x1N, x21, ... , xCD ]
>
> When you ask:
>
> "Do you know of any calculus tricks to get the probabilities over every
> possible candidate position, or is this simply too hairy?"
>
> You are asking, "how do I integrate f()"? Your default plan of drawing
> random candidates and computing f() for each is integration by Monte Carlo
> sampling. Now, I don't know how to integrate f() analytically, but
> realizing that it is an integral allows us to consider other forms of
> numerical integration (hence my request that f() be a smooth function). One
> such example is Laplace's method:
>
> https://en.wikipedia.org/wiki/Laplace%27s_method
>
> Let us imagine that f() looks like a hill. If you can compute the gradient
> of f(), you can climb the hill and get to the peak. Up there, if you can
> compute the Hessian of f(), you can approximate f() with the Gaussian
> function with the same height and Hessian. This method is approximate, but
> it is very fast because you do not have to sample across the entire space.
> Laplace's method is often used in highly dimensional problems, where the
> cost of random sampling is prohibitive.
>
> How do you compute the derivative and Hessian? Most often this is used
> with an Autodiff package. Alternatively, you can use finite differences but
> at the cost of reduced accuracy. What if f() isn't one hill but many? You
> can integrate each hill separately and add them up.
>
> Anyway, I don't know whether Laplace's method is a realistic solution, but
> if it isn't, perhaps there is a different form of numerical integration
> tool that is available to you that is more efficient than Monte Carlo
> integration. For example, importance sampling doesn't require derivatives.
> There is a long list of numerical integrators that have been devised over
> the ages.
>
> Cheers,
> --
> Dr. Daniel Carrera
> Postdoctoral Research Associate
> Iowa State University
> ----
> Election-Methods mailing list - see https://electorama.com/em for list
> info
>
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