[EM] Exact spatial model probabilities?

[Daniel Carrera]

On Wed, Jan 26, 2022 at 10:19 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> Practical suggestions ... convert the second order vector differential
> equations into a system of two first order vector ODE's ...
>
> dr/dt=v,
> dv/dt=-grad V(r)
>
> Solve numerically using RK4 or even simple Euler's method ... after all
> we're just using the path to sample the space. Try several initial
> conditions ... nothing too far outside the support of the voter
> distribution.
>

I'm pretty sure that that would just create an orbit. Since V(r) is not a
point mass we might expect a complicated non-periodic or quasi-periodic
orbit, but an orbit nonetheless. If we knew that V(r) had a simple bowl
shape we could just do a hill-climbing algorithm, but in general V(r) could
have all sorts of hills, valleys, and saddle points.

One simple example: If you have a cluster of left-wing voters and a cluster
of right-wing voters, and they are well segregated, you could end up just
orbiting one of the two clusters. That could say something interesting
about why it might be hard for a politician to be a centrist (politician
can't climb out of her respective local potential to move to the center)
but that's not what you are looking for.

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