[EM] Exact spatial model probabilities?

[Forest Simmons]

Just one correction ****

El mié., 26 de ene. de 2022 10:02 a. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

>
>
> 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)
>
*** Should be
(d/dt)^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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