Anonymous at Mon, 17 Feb 2025 10:49:18 UTC No. 16589068

this nigga doesn't know about the central limit theorem

Anonymous at Mon, 17 Feb 2025 10:50:16 UTC No. 16589069

>>16589068
>NPCs are a math theorem

Anonymous at Mon, 17 Feb 2025 11:31:41 UTC No. 16589107

>>16589066
It cannot

Anonymous at Mon, 17 Feb 2025 12:10:22 UTC No. 16589138

>>16589069
G-D said adding random distributions together gives you Gauss's distribution (pbuh) and G-D is always right

Anonymous at Tue, 18 Feb 2025 01:55:59 UTC No. 16589764

>>16589068
and you dont know about limits

Anonymous at Tue, 18 Feb 2025 04:03:22 UTC No. 16589834

>>16589066
I don't think anyone who actually knows anything about probability thinks that.

People use Gaussian distributions for 3 reasons:
1) They are computationally super simple to deal with (any Gaussian random variable can be created by a linear transformation of a zero-mean unity variance Gaussian).
2) They are usually a good enough approximation locally. If you are measuring a statistic which takes a sum or an average of a bunch of independent quantities (and some kinds of correlated quantities depending on the correlation structure), you can likely get away with using an appropriately scaled/located Gaussian as an approximation of the "true distribution."
3) Gaussians are entirely characterized by their first 2 moments. A lot of people talk about the "fat tails" problem for leptokurtic distributions. That's really mostly an academic issue. The big thing that Gaussians have going for them is that "uncorrelated" and "independent" are the same thing for two Gaussians. This is not in general true, and you can have higher order or non-linear dependencies which will make your life bad if things are not Gaussian. This isn't a problem if you shrug your shoulders and say "we assume it's a Gaussian."

Anonymous at Tue, 18 Feb 2025 08:26:48 UTC No. 16589940

>>16589834
r u a smart anon?

Anonymous at Tue, 18 Feb 2025 22:40:07 UTC No. 16590812

>>16589138
fun fact if you add distributions with infinite variance and finite mean you will get non-Gaussian distributions

Anonymous at Wed, 19 Feb 2025 02:51:58 UTC No. 16591028

>>16589940
Nah, I'm dumb as a bag of rocks. I just have spent some time studying math/EE. Convergence is actually pretty cool once you understand what's happening.

Anonymous at Wed, 19 Feb 2025 14:19:35 UTC No. 16591429

>>16589834
>noooo we use it bcs its usefulllll!!!!111
>BTW way too retarded to solve this
KYS FAG

Anonymous at Wed, 19 Feb 2025 18:39:38 UTC No. 16591672

>>16591429
this isn't /b/

Anonymous at Wed, 19 Feb 2025 18:44:30 UTC No. 16591677

>>16591429
What do pdes have to do with whether or not Gaussian approximations are reasonable? Do you have brain worms?

Anonymous at Wed, 19 Feb 2025 18:52:09 UTC No. 16591693

>>16591677
the heat equation is solved by integrating a gaussian over the initial condition (or if you start with a point it's just a gaussian)
that said, yes he does seem to have brain worms

Anonymous at Wed, 19 Feb 2025 18:56:39 UTC No. 16591698

>>16591693
Anything can be solved by convoluting over some exponential function.

Anonymous at Wed, 19 Feb 2025 18:59:02 UTC No. 16591699

>>16591698
yeah anything elliptic, maybe?
the heat equation is the one that corresponds to brownian motion and so a zero mean gaussian

Anonymous at Thu, 20 Feb 2025 01:00:39 UTC No. 16592012

>>16589066
can it really?, it seems so. I am studying applied math and it seems like the entire program has been one giant argument for why everything is ultimately a stochastic differential equation.
it's a nice framework to build on, you can define your mean term and your variance term to do whatever you like, you can add dynamics between them, define multivariate variables with covariance, shit's pretty wild and interesting, thought I don't see how any of this is useful (other than finance and perhaps other forms of gambling)

Anonymous at Thu, 20 Feb 2025 01:34:36 UTC No. 16592041

>>16592012
The entire AI and deep learning thing is based on probability theory.

Anonymous at Thu, 20 Feb 2025 01:55:55 UTC No. 16592060

>>16591693
Interesting. I've never done a PDEs course, nor needed them in any real capacity. I use probability and Fourier analysis on WSS stochastic processes daily for work, but never really end up needing to deal with stochastic PDEs. Most of my research areas tend to involve deterministic ergodic ODE systems excited by bandpass Gaussian processes.

Anonymous at Fri, 21 Feb 2025 20:58:38 UTC No. 16595435

>>16589834
> A lot of people talk about the "fat tails" problem for leptokurtic distributions. That's really mostly an academic issue.

This nigga clearly does not know anything about non-file insurance contract pricing or solvency model for reinsurance companies.

Anonymous at Sat, 22 Feb 2025 05:01:51 UTC No. 16595935

>>16595435
I don't know much about insurance, but I do know one for certain. You need to stop reading Taleb.

The guy is confusing two distinct ideas and just clumps them together under "fat tails." Firstly there are properly leptokurtic distributions like higher order alpha-stable distributions or Cauchy distributions. These generally show up not because of anything actually fundamental about the specific underlying phenomenon, but actually because the "independent samples" they use for getting a distributional fit for their problem actually are correlated time-series realizations.

The second kind of "fat tails" problem, which is pretty much the entire focus of the black swan, despite being mislabeled, is actually a mixture model problem. If you have a distribution which is a mixture and is multimodal, it will be fat tailed if you're only looking at excess kurtosis.

That isn't because of the "high probability of rare events," it's because kurtosis is only fundamentally meanful as a measure of rate decay for unimodal distributions. You could have two incredibly tight Gaussians in a mixture and your sample kurtosis will say "fat tails" if you just chuck a numerical expectation at it and don't use your brain.

Anonymous at Sat, 22 Feb 2025 05:19:26 UTC No. 16595950

>>16589066
>Lies
>Damn Lies
>Statistics

Anonymous at Sat, 22 Feb 2025 12:12:11 UTC No. 16596250

>>16595935
>Firstly there are properly leptokurtic distributions like higher order alpha-stable distributions or Cauchy distributions. These generally show up not because of anything actually fundamental about the specific underlying phenomenon, but actually because the "independent samples" they use for getting a distributional fit for their problem actually are correlated time-series realizations.

That does not change the fact that the observed sample will look like something drawn form a fat tailed distribution. If you are a reinsurance company, say re-insuring companies insuring properties in Los Angeles, of course you are aware that losses from individual companies, due to a wildfire event, will be correlated. But if your overall loss could as well modeled with an Pareto distribution, why not go with the fat tailed distribution?

Anonymous at Sat, 22 Feb 2025 16:40:35 UTC No. 16596484

>>16596250
> But if your overall loss could as well modeled with an Pareto distribution, why not go with the fat tailed distribution?

If you have a better model than a Gaussian for your particular use-case, then there's no problem. In my world (acoustic signal processing) people use Cauchy noise models all the time to approximate the impacts of long-lasting but low-power correlations.

The point I was making with the "it's mostly an academic issue" is that you can usually get away with a Gaussian.

If you are in one of the rare circumstances where your particular niche has a better model to use, use that. Otherwise, a Gaussian is a good first guess and has a lot of justifiable properties.

These are all models anyways. They are by definition, wrong. The point is which models are useful. If your use case has a more useful model and the computational benefits you lose for not leveraging Gaussianity don't matter, just use that model.

Anonymous at Sat, 22 Feb 2025 16:56:20 UTC No. 16596501

>>16596484
What randomness in nature that has one mode, absolutely symmetrical around that mode and doesn't suffer from one or more irregularities at the tails? I honestly don't understand why the Guassian works at all.

Anonymous at Sat, 22 Feb 2025 17:02:30 UTC No. 16596506

>>16596501
it's not hard in nature at all to find a large sum of independent variables
sure there's always "irregularities at the tails" but often you have to go so far into the tail that the Gaussian is practically zero

Anonymous at Sat, 22 Feb 2025 17:07:33 UTC No. 16596511

>>16596506
So the Guassian is some generic easy approximation function of an unknown much harder to obtain pdf that actually describes your specific model?

Anonymous at Sat, 22 Feb 2025 17:10:08 UTC No. 16596514

>>16596511
sure yeah
but you do have central limit theorems to say it's an arbitrarily good approximation to a large sum
and even berry-esseen theorems to control how close it is

Anonymous at Sat, 22 Feb 2025 17:13:34 UTC No. 16596520

>>16596514
I really need to dive into that stuff mathematically, everybody just assumes that the Guassian can represent almost distribution (e.g. IQ, wealth, height, etc...) but I still can't understand why it is so ubiquitous, it just doesn't make any sense that so many unrelated random things imitate this equation for some mysterious reason.

Anonymous at Sat, 22 Feb 2025 17:18:43 UTC No. 16596525

>>16596520
>IQ, wealth, height, etc...
yeah it's not obvious why these should be gaussian at all (and I think wealth is not gaussian actually)
you could maybe think of a ridiculous model for height where each small part of your body has a random length
and if we believe that then you've explained gaussian height

Anonymous at Sat, 22 Feb 2025 17:25:25 UTC No. 16596529

>>16596525
Yeah sorry, wealth follows Pareto and its similar functions. But I meant that so many unrelated things tend to follow the Guassian which is fn(e^-x^2) so why not fn(e^-x^6) for example or why e as the base at all? what's so special about e^-x^2?

Anonymous at Sat, 22 Feb 2025 17:28:29 UTC No. 16596533

>>16596529
that's just what big sums of finite variance terms look like idk
not really got a better explanation than >>16589138

Anonymous at Sat, 22 Feb 2025 20:00:28 UTC No. 16596678

>>16596529
This guy here: >>16596484

The answer for "why e^(-x^2)?" comes down to two basic things. Firstly, for probability distributions with finite mean [math]\mu[/math] and finite variance [math]\sigma^2[/math], the Gaussian distribution maximizes the entropy on the reals.
[math]
p(x) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} = \arg_{p \in \mathcal{L}^2(\mathbb{R},\lambda)}\max_{\mu, \sigma^2} h(X)
[/math]

What this means in practice is that of distributions on the real numbers with a finite mean and finite variance, the Gaussian has the maximum possible "internal disorder."

So when you're taking a sample mean of i.i.d. samples, that sample mean converging to a Gaussian is telling you it is maximally internally disordered.

The central limit theorem actually is a good bit more general than the Gaussian case. Basically your sum over a particular domain always converges to the maximum entropy (a.k.a. maximally disordered) distribution on that space. If you were looking to approximate to the 4th order, you'd get a 4th order exponential distribution (it one exists for the underlying moments).

Anonymous at Sat, 22 Feb 2025 20:03:01 UTC No. 16596684

>>16596678
>>16596529

Secondly, the Gaussian distribution is the highest order bounded variation distribution on the reals for which the maximum entropy distribution always exists. When you have a maximum entropy distribution with non-trivial skew or kurtosis, you're not even guaranteed to have a convergent distribution (a.k.a. your law of large numbers based sum might actually just not converge for your case).

These two combined with the fact that mean and variance are usually enough to get a good model of how something works makes the Gaussian a no-brainer unless you know for certain you have skew or excess kurtosis.

raphael at Sat, 22 Feb 2025 20:07:46 UTC No. 16596689

>>16589138
shut up kike

Anonymous at Sat, 22 Feb 2025 21:48:56 UTC No. 16596794

>>16596678
>The central limit theorem actually is a good bit more general than the Gaussian case. Basically your sum over a particular domain always converges to the maximum entropy (a.k.a. maximally disordered) distribution on that space. If you were looking to approximate to the 4th order, you'd get a 4th order exponential distribution (it one exists for the underlying moments).
this sounds interesting but what is a 4th order exponential distribution?
the only other limits I know are of power laws with some fixed largest moment (less than 2)

Anonymous at Sat, 22 Feb 2025 22:07:23 UTC No. 16596805

>>16596678
>>16596794
oh I partially figured it out, you mean the exponential family where the log density is a quartic
but don't these variables have a finite second moment and so sum to a gaussian?

🗑️ Anonymous at Sun, 23 Feb 2025 02:07:48 UTC No. 16596987

>>16596805
I was being slightly imprecise and sloppy with my wording and said something that's actually not quite correct but is somewhat close.

You could look up the generalized Central Limit Theorem if you'd like, but it's a bit opaque.

Basically the idea is that if you have a sequence:
[math]\{a_n: n\in \mathbb{N}, a_n >0 \forall n\}
[/math]
and a sequence
[math]
{b_n: n\in \mathbb{N}, b_n\in \mathbb{R}\}
[/math]
with i.i.d. samples [math]{X_i}_{I=1}^{N}[/math] then if the sequence:
[math]
a_n \Sum_{i=1}^{n}X_i -b_n \to_{d} Z
[/math]
we call Z a "stable" or "alpha-stable" distribution.

The Gaussian case has [math]a_n = 1/n[/math] and [math]b_n = \mathbb{E}[X_1][\math]. However, this convergence to specifically the Gaussian only happens because as [math]n\to\infty[/math], the set of i.i.d. realizations [math]\{X_i\}[/math] becomes (with probability 1) maximally disordered on its support.

So you are right that the 4-parameter exponential family distribution should have the errors of its sample-mean converge to a Gaussian distribution.

However, for other statistics and other sample spaces you'd see convergence of errors to other maximum entropy distributions.

For example, if you are looking at the sample mean error for a directional distribution on a unit sphere, this will converge to a von-Mises distribution (which does look an awful lot like a Gaussian locally, so it's usually okay to approximate by a Gaussian if you don't want to deal with your sample space being a Riemannian sub-manifold).

Similarly, if you estimate the sample median of i.i.d. samples the errors converge to a Laplace distribution.

It's basically just about the particular statistic being used, and the general idea of "spreading out as much as possible" (which is what the maximum entropy part of things is doing).

Anonymous at Sun, 23 Feb 2025 02:17:20 UTC No. 16597001

>>16596805
I was being slightly imprecise and sloppy with my wording and said something that's actually not quite correct but is somewhat close.

You could look up the generalized Central Limit Theorem if you'd like, but it's a bit opaque.

Basically the idea is that if you have a sequence:
[math]
\{a_n: n\in \mathbb{N}, a_n >0\forall n\}
[/math]
and a sequence
[math]
\{b_n: n\in \mathbb{N}, b_n\in \mathbb{R}\}[/math]
with i.i.d. samples [math]X_1,X_2,...[/math]
then if the sequence:
[math]
a_n \sum_{i=1}^{n}X_i−b_n \to_d Z
[/math]
we call Z a "stable" or "alpha-stable" distribution.

The Gaussian case has
[math]
a_n=1/n
[/math]
and
[math]
b_n=\mathbv{E}[X_1]
[/math].

However, this convergence to sp
ecifically the Gaussian only happens because as
[math]n\to \infty [/math], the set of i.i.d. realizations {Xi} becomes (with probability 1) maximally disordered on its information space.

So you are right that the 4-parameter exponential family distribution should have the errors of its sample-mean converge to a Gaussian distribution.

However, for other statistics and other sample spaces you'd see convergence of errors to other maximum entropy distributions.

For example, if you are looking at the sample mean error for a directional distribution on a unit sphere, this will converge to a von-Mises distribution (which does look an awful lot like a Gaussian locally, so it's usually okay to approximate by a Gaussian if you don't want to deal with your sample space being a Riemannian sub-manifold).

Similarly, if you estimate the sample median of i.i.d. samples the errors converge to a Laplace distribution.

It's basically just about the particular statistic being used, and the general idea of "spreading out as much as possible" (which is what the maximum entropy part of things is doing)

🗑️ Anonymous at Sun, 23 Feb 2025 03:12:10 UTC No. 16597037

>>16597001
Oops. The errors of sample median absolute deviation will converge to a Laplace distribution. Not the sample median itself.

Anonymous at Sun, 23 Feb 2025 03:14:08 UTC No. 16597038

>>16597001
Oops. The errors of the sample mean absolute deviation converge to a Laplace distribution. Not the errors of the sample median directly. Late night phone posting, not even once.