[EM] Exact spatial model probabilities?

[Forest Simmons]

So we can approximate voter distributions with sums, products,
convolutions, etc of standard library distributions. And then get the mean,
variance, and other moments via MacLauren series coefficients ...AND vice
versa ... if we know the desired moments, we can construct the MacLauren
series for the transform of the distribution function, and then inverse
trans form back to the desired distribution function. Classically this was
called. "The problem of the moments."

The biggest difficulty was dealing with  divergent or slowly convergent
MacLauren series ... the solution was to use continued fraction expansions
or other Pade approximants to get convergence. Nowadays numerical inversion
of these transforms is much more practical than it was in those days.

Of course all of this is trivial in one dimension ... but not in two or
three dimensions.

At least an answer to Colin's question about the proper or central
mathematical setting for voting methods is starting to take shape.

(She's taking off again on a long errand of mercy with the phone ... so
more later)

El mar., 25 de ene. de 2022 12:10 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> To paraphrase Dickens ... they were the good old days ... and the bad old
> days!
>
> Daniel's idea reminds me of Robert  B-J's comment about Heaviside and
> Dirac  functions. Dirac unit impulse functions are approximated by
> Gaussians with infinitesimal variance in the Theory of Distributions. And
> every Probability distribution is a convolution of itself with a Dirac
> delta ... which is useful because Laplace transforms turn convolution
> products into algebraic products.
>
> Electrical engineers are used to approximating all kinds of input signals
> with sums of standard functions ... impulse, step, ramp, sinusoids,
> Gaussians, that have well known Laplace and Fourier Transforms.
>
> How useful it is to be able to go back and forth between the time and
> frequency domains! Even in quantum mechanics ... the more compact the
> support of a wave function, the more spread out its Fourier transform, and
> vice-versa. That's the wave mechanical basis of the uncertainty principle.
>
> A convenient way to get the mean, variance, and higher moments of a
> probability distribution (think voter distribution) is by finding the
> Taylor/MacLauren coefficients of the Laplace or Fourier transforms of the
> distribution functions.
>
> [Fourier and Laplace transforms differ by a 90 degree rotation in the
> complex frequency domain.So what I say for one also goes for the other
> without needing to mention it every time.]
>
> My wife needs the phone ... more later..
>
> W
>
> El mar., 25 de ene. de 2022 9:40 a. m., Kristofer Munsterhjelm <
> km_elmet at t-online.de> escribió:
>
>> On 25.01.2022 06:06, Forest Simmons wrote:
>> > Thanks Ted and Daniel. Very interesting!
>> >
>> > In the early 70's we did our minuteman missile simulations on a room
>> > size mainframe IBM 360/65 computer with FORTRAN code, double precision
>> > arithmetic ... punched cards interface.... and all. We got one
>> > turnaround per night.... night because the Top Secret runs had to be
>> > totally isolated from the daytime use of the computer. In 1974 our group
>> > got ahold of a couple of the mini-computers that were just coming out
>> > ... TTY "ticker tape" interface at first then (unreliable, but more
>> > convenient) floppy discs. Very slow, single precision, but interactive
>> > BASIC for the spine of the simulation. We employed pseudo-double
>> > precision for the numerical integration, and modified BASIC so we could
>> > call on assembled bottle-neck subroutines, etc.
>>
>> It's things like these that makes me think that current computers are
>> capable of vastly more than they're currently being used for. Computers
>> with less than 1M of RAM could be used to calculate missile
>> trajectories, run industrial process control, etc. We now have 16G or
>> more.
>>
>> Of course, I know that part of the reason is that programmer time is now
>> the most scarce resource. The programs that are developed now (mostly
>> user-facing stuff) are much slower than they need to be in part because
>> it would take too much time and effort to optimize down to the bare metal.
>>
>> -km
>>
>
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