[EM] Exact spatial model probabilities?

[Forest Simmons]

El mié., 26 de ene. de 2022 3:12 a. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> On 26.01.2022 04:23, Forest Simmons wrote:
>
> > The dirac delta is the convolution identity distribution ... convolving
> > it with another distribution leaves it unchanged with cliffs and sharp
> > corners intact.
> >
> > But if you convolve with a smooth approximation of Dirac,  like a
> > gaussian with tiny variance, you get an infinitely differentiable
> > approximation of the "horrible" function.
>
> Right. I once wrote a fully differentiable genetic algorithm (I was
> intending to use it for hyperparameter tuning). There were two problems.
> First, local optima. Second, suppose that you have a plain statement like:
>
> if (x>y) {
>         return z;
> } else {
>         return f(z);
> }
>
> When smoothing, this turns into something like
>
> return z * sig(x-y, k) + f(z) * sig(y-x, k)
>
> where sig(x, k) is a sigmoid function that returns 1 if x>>0, 0 if x<<0,
> and k is a steepness parameter controlling sig'(x) around 0.
>
> If k is too high, then gradient descent fails because there's no
> noticeable slope down to the optimum; it's flat almost everywhere and
> then goes to zero exactly at the optimum (vanishing gradient problem).
>
> So we need some smoothness, which means that both branch values come
> into play. But this makes the function slow to evaluate as the program
> must "go both ways" on every conditional.
>

Here's a better way to get a smooth potential function V(x,y,z,) in the
present context:
First the potential V(x,y,z)for all voters concentrated at one point
(X,Y,Z):

V(r)=||r-R||, where r=(x,y,z) and R=(X,Y,Z).

To smooth, replace this with

V(r)=(epsilon+||r-R||^2)^(1/2)

This is more realistic anyway because you cannot truly have all voters
concentrated at one point. The epsilon smear factor would be a function of
the variance of the voters clustered around the point R.

That variance could be given exactly for a uniform distribution on a ball
of radius delta centered at R, for example.

Then for multiple factions...

V(r)=Sum(over R in Omega) of
f(R)*sqrt(epsilon(R)+||R-r||^.5),

where Omega is the set of faction centers, and f(R) is the fraction of
voters clustered near R, i.e. in a delta(R) [related to epsilon(R)]
neighborhood of R.

If we have a smooth (i.e. pre-smeared) pdf f(R), we can write the
"electo-potential" V(r) as an integral
V(r)=Integral(over R in Omega)of
               ||r-R||f(r)dxdydz

The hamiltonian for the wandering voter particle is the total energy,
kinetic + potential, E=T+V, where T =.5 ||dR/dt||^2.

The system (in vector form) of ODE's for the motion of the particle is

(d/dr)^2=-grad V(r)

I need to check the HMC link that Daniel gave me to see what notation they
are using.

>
> > Electrical engineers have a vast library of standard test patterns to
> > use as input signals for use in designing and testing circuits.
> >
> > We need a similar library of test distributions for use in designing and
> > testing election methods.
> >
> > Election methods could even be profiled by their responses to these test
> > patterns.
>
> This, I do agree with. There have been a lot of voting method proposals
> lately, and we need some way to easily determine:
>
> - what is its VSE (under what models)
> - what is its voter-strategy susceptibility
> - what is its candidate-strategy susceptibility (cloning)
> - what criteria does it definitely fail
>
> and it would be nice to also know what criteria it definitely passes,
> though that requires formal verification, which is much harder than just
> testing a bunch of cases.
>
> (I remember using REDLOG to come up with BTV once, but I don't remember
> the details. Perhaps I should look into the current state of the art for
> provers, like Z3... so many things to do and so little time in which to
> do them!)
>
> -km
>
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