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The math that explains why bell curves are everywhere

quantamagazine.org

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Re: The math that explains why bell curves are everywhere

#111

Post by AlexCornila »

mikrl wrote: post #3

Great article. Personally I have been learning more about the mathematics of beyond-CLT scenarios (fat tails, infinite variance etc) The great philosophical question is why CLT applies so universally. The article explains it well as a consequence of the averaging process. Alternatively, I’ve read that natural processes tend to exhibit Gaussian behaviour because there is a tendency towards equilibrium: forces, homeost…

that’s correct but a better explanation is this https://youtu.be/AwEaHCjgeXk?si=tV72uauquCHvzkNE

Re: The math that explains why bell curves are everywhere

#112

Post by trymas »

bux93 wrote: post #108

The article doesn't share the actual math, but also not the relatively easy intuition. When you roll a pair of dice, there are more combinations that add up to 7 than any other number. Change the numbers on the dice (change the 1 to a 6, e.g.), there's again more combinations that add up to some numbers than to others. The histogram of the number of combinations that add up to different results is a bell curve. That'…

> It's sad that even introductory statistics courses skip this simple intuition.

I was probably lucky.

We got homework as one of the first lessons in statistics course, for exactly this case.

Roll pair of dice, save the result, do it 200 (or some other bigger number) times, plot the histogram, do some maths, maybe provide any conclusions, etc.

Such things then definitely stuck with you for a long time.

Re: The math that explains why bell curves are everywhere

#113

Post by UncleSlacky »

CoastalCoder wrote: post #81
srean wrote: post #80

The way I understand this is that adding of random variables is a smoothening operation on their densities (more generally the distributions, but let me speak of densities only). A little more formally, additions over random variables are convolutions of their densities. Repeated additions are repeated convolutions. A single convolution can be understood as a matrix multiplication by a specific symmetric matrix. Repe…

> Anyone familiar with linear algebra will know that repeated matrix multiplication by non degenerate matrices reveals it's eigenvectors. TIL that I'm not "familiar" with linear algebra ;) But seriously, thanks for sharing that knowledge.

Obligatory XKCD: https://xkcd.com/2501/

Re: The math that explains why bell curves are everywhere

#115

Post by estearum »

AlexCornila wrote: post #109
bux93 wrote: post #108

The article doesn't share the actual math, but also not the relatively easy intuition. When you roll a pair of dice, there are more combinations that add up to 7 than any other number. Change the numbers on the dice (change the 1 to a 6, e.g.), there's again more combinations that add up to some numbers than to others. The histogram of the number of combinations that add up to different results is a bell curve. That'…

Yeah that is definitely not the relatively easy intuition for this. The relatively easy intuition comes from learning about the Bernoulli trials, binomial distribution and Pascal triangle. Once you understand those you understand why normal distribution is so prevalent. Or just watch this https://youtu.be/AwEaHCjgeXk?si=tV72uauquCHvzkNE

They explained the intuition in like two sentences that an 8th grader can understand and test themselves.

Sounds simpler than whatever you’re talking about here

Re: The math that explains why bell curves are everywhere

#116

Post by srean »

bux93 wrote: post #108

The article doesn't share the actual math, but also not the relatively easy intuition. When you roll a pair of dice, there are more combinations that add up to 7 than any other number. Change the numbers on the dice (change the 1 to a 6, e.g.), there's again more combinations that add up to some numbers than to others. The histogram of the number of combinations that add up to different results is a bell curve. That'…

And if you take the log of the number of ways you get the entropy corresponding to the 'ways' macrostate.

Re: The math that explains why bell curves are everywhere

#117

Post by fritzo »

orangemaen wrote: post #16
fritzo wrote: post #7

Hot take: bell curves are everywhere exactly because the math is simple. The causal chain is: the math is simple -> teachers teach simple things -> students learn what they're taught -> we see the world in terms of concepts we've learned. The central limit theorem generalizes beyond simple math to hard math: Levy alpha stable distributions when variance is not finite, the Fisher-Tippett-Gnedenko theorem and Gumbel/Fr…

The CLT is everywhere because convolution/adding independentish random variables is a super common thing to do.

Right. And the CLT is not actually limited to normal distributions. Both of the distribution families I mentioned are central limit theorems. The CLT we first see in school regards means of finite variance distributions, where the finite variance assumption is made because it makes the math easier.

https://en.wikipedia.org/wiki/Central_limit_theorem#The_gene...

Re: The math that explains why bell curves are everywhere

#118

Post by fritzo »

gowld wrote: post #24
fritzo wrote: post #7

Hot take: bell curves are everywhere exactly because the math is simple. The causal chain is: the math is simple -> teachers teach simple things -> students learn what they're taught -> we see the world in terms of concepts we've learned. The central limit theorem generalizes beyond simple math to hard math: Levy alpha stable distributions when variance is not finite, the Fisher-Tippett-Gnedenko theorem and Gumbel/Fr…

Most things aren't infinite or extreme, though. Almost by definition, most phenomena aren't extreme phenomena.

Heavy tails are everywhere. Normal distributions have absurdly light tails. Levy alpha stable distributions have power law tails. Power law tails are everywhere.

Some things with heavy tails:

  token occurrences
  comment thread upvotes
  startup IPOs
  social follower counts
  network latency
  github stars
  git diffs
  power station size
  weather events

Re: The math that explains why bell curves are everywhere

#119

Post by fritzo »

roenxi wrote: post #78
BobbyTables2 wrote: post #14

Earlier quoted context omitted.

It also took me a little while to realize “least squares” and MMSE approaches were not necessarily the “correct” way to do things but just “one thing we actually know how to do” because everything else is much harder. We can use Calculus to do so much but also so little…

That isn't the case; mathematicians will do pages of calculations (particularly and especially the statisticians) if they can prove one approach is technically superior to another. These people, as a class, are the crazies who invented matrix multiplication. Something like MMSE is used because it provably optimum properties for estimating a posterior distribution. It is certainly possible that there are complex appro…

> optimum properties for estimating a posterior distribution

Circular reasoning: that's true only if the posterior is normal, or if your "optimal" is defined by second moments. In infinite variance cases, the best estimator can be median or an alpha moment for alpha -- A mathematician who has indeed fallen into the beauty trap

Re: The math that explains why bell curves are everywhere

#120

Post by cortesoft »

bux93 wrote: post #108

The article doesn't share the actual math, but also not the relatively easy intuition. When you roll a pair of dice, there are more combinations that add up to 7 than any other number. Change the numbers on the dice (change the 1 to a 6, e.g.), there's again more combinations that add up to some numbers than to others. The histogram of the number of combinations that add up to different results is a bell curve. That'…

Anybody who has ever played Craps will know this.
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