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The Man Who Invented Modern Probability

nautil.us

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Re: The Man Who Invented Modern Probability

#31

Post by nawitus »

mag487 wrote: post #29
nawitus wrote: post #21

Earlier quoted context omitted.

Additional question: what's the Kolmogorov complexity of a irrational numberwhich is not described with a ZFC formula? And how is it described?

I haven't really studied algorithmic information theory, but I'd assume that Kolmogorov complexity isn't defined for uncomputable/undefinable numbers. Or maybe it's defined as "infinite," but either way, my guess is that such numbers are simply ignored. (Interestingly, the function which takes a computable number and outputs its Kolmogorov complexity is itself uncomputable!)

The article at least implies that the Kolmogorov complexity of uncomputable irrational numbers is larger than computable irrational numbers.

Re: The Man Who Invented Modern Probability

#32

Post by SatvikBeri »

tzs wrote: post #30
SatvikBeri wrote: post #22

Earlier quoted context omitted.

If the side length isn't constrained to integers, the expected volume is 39 inches cubed. In a bit more detail: take the integral of x^3 (the volume function) from 1 to 5 and you get 5^4/4 - 1^4/4 = 624/4 = 156. Then we divide by the length of the line segment (4) to get 39. If the side lengths have to be integers, then it's just (1 + 8 + 27 + 64 + 125)/5 = 45 inches. Edit: corrected division mistake, thanks to dxbyd…

> Edit: corrected division mistake, thanks to dxbydt Well, one of the mistakes: > If the side lengths have to be integers, then it's just (1 + 8 + 27 + 125)/4 = 40.25 inches If I'm not mistaken, 4 is an integer, and so should not have cruelly snubbed when you sent out invitations to your summation of the cubes of the integers from 1 to 5. Sure, 4 is not a cool integer like the others in 1 to 5...he's got a reputation…

Oy. Someone needs to revoke my math license. Thanks for the correction!

Re: The Man Who Invented Modern Probability

#33

Post by dxbydt »

SatvikBeri wrote: post #32
tzs wrote: post #30

Earlier quoted context omitted.

> Edit: corrected division mistake, thanks to dxbydt Well, one of the mistakes: > If the side lengths have to be integers, then it's just (1 + 8 + 27 + 125)/4 = 40.25 inches If I'm not mistaken, 4 is an integer, and so should not have cruelly snubbed when you sent out invitations to your summation of the cubes of the integers from 1 to 5. Sure, 4 is not a cool integer like the others in 1 to 5...he's got a reputation…

Oy. Someone needs to revoke my math license. Thanks for the correction!

My, my, 2 mistakes on a silly uniform distribution! You must atone for this sin. Explain what the volume would be if the side came from a normal distribution over [1,5] (ie. mean 3, sigma 2/3) Otherwise I will personally call the MAA director on my speed dial & haul your ass back to stat101 :)

But seriously, you write rather well on Quora & I'm sure the crowd would want to know what happens to the volume were you to choose a normal distribution instead of a uniform, so I hope you respond.

Re: The Man Who Invented Modern Probability

#34

Post by nwhitehead »

nawitus wrote: post #20
nwhitehead wrote: post #14

Earlier quoted context omitted.

Consider the set P of programs that take no input and generate an infinite stream of digits. Each program in P has a finite length and is written out of a finite set of symbols, so there must only be a finite number of programs of any given length. That makes P countable. Let Q be the set of numbers described by programs in P. Each program from P describes exactly one number, so Q must also be countable. The set of i…

Your proof seems to only prove that a finite number of programs (described by you) which can produce a finite number of irrational numbers, while there are infinite number of irrational numbers. But we're surely not talking about a finite number of programs. I'm not even sure what the Kolmogorov complexity of a "complex irrational number" means. If you need the sequence of digits and you cannot use an algorithm to pr…

For each fixed length there are a finite number of programs of that length. If we use 8-bit bytes for the alphabet, there are 256^N programs of length N. The set of ALL programs is infinite (we don't limit the length of programs), but it is countable.

The "countable" part means we can put the set into one-to-one correspondence with the natural numbers. The correspondence starts with 0 mapping to the empty program, then 1-256 mapping to programs of a single byte, then 257-65793 mapping to the programs of two bytes, and so on. This mapping will hit each program exactly once, and it will hit every program because every individual program has a finite length.

Different types of infinities are not intuitive so don't feel bad if the concepts are confusing. These issues troubled lots of very smart mathematicians for decades. The existence of irrational numbers was hugely troubling to Pythagoreans. The existence of uncountable infinities discovered by Cantor was shocking [1].

[1]: http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_the...

Re: The Man Who Invented Modern Probability

#36

Post by lutusp »

tlarkworthy wrote: post #35

Discovered, not invented.

> Discovered, not invented.

This refers to a philosophical debate that asks whether mathematics is part of nature or an artificial invention of man. I think the debate has begun to lean toward mathematics being part of nature, in which case yes, the proper term is discovered.

Re: The Man Who Invented Modern Probability

#39

Post by wamatt »

te_platt wrote: post #2

An interesting take the Kolmogorov system can be found here: http://www-biba.inrialpes.fr/Jaynes/cappal.pdf In it E. T. Jaynes compares Bayesian statistics Kolmogorov probabilities and finds them essentially identical in result even after working from very different first principles.

For anyone interested, the above appendix comes from ET Jaynes's[1] magnum opus, "Probability Theory: The Logic of Science" [2].

Which, IMHO is deserving of it's grand title and provides an excellent framework for using Bayes in the real world.

[1] http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes

[2] http://shawnslayton.com/open/Probability%20book/book.pdf

Re: The Man Who Invented Modern Probability

#40

Post by Cogito »

dxbydt wrote: post #33
SatvikBeri wrote: post #32

Earlier quoted context omitted.

Oy. Someone needs to revoke my math license. Thanks for the correction!

My, my, 2 mistakes on a silly uniform distribution! You must atone for this sin. Explain what the volume would be if the side came from a normal distribution over [1,5] (ie. mean 3, sigma 2/3) Otherwise I will personally call the MAA director on my speed dial & haul your ass back to stat101 :) But seriously, you write rather well on Quora & I'm sure the crowd would want to know what happens to the volume were you to…

From [0]:

Let X be a random variable and g be any function.

If X is continuous, then the expectation of g(X) is defined as,

E[g(X)] = \int_−∞^∞ g(x)f(x) dx,

where f is the probability density function of X.

---

For us, with mean 3 and sigma 2/3, f = 3/2 e^(-9/8 (x-3)^2)/sqrt(2 π)

We need to integrate numerically; using Wolfram [1] we get an estimate of 30.8099.

If we ask for the expected value over the entire distribution we get 31.

[0] http://imai.princeton.edu/teaching/files/Expectation.pdf

[1] http://www.wolframalpha.com/input/?i=integrate+3%2F2+e%5E%28...

[2] http://www.wolframalpha.com/input/?i=expected+value+of+x%5E3...

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