The Man Who Invented Modern Probability
1–10 of 73 posts
Re: The Man Who Invented Modern Probability
#2An 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.
Re: The Man Who Invented Modern Probability
#3My father was the grand-grad student of Kolmogorov. He (Kolmogorov) is a very impressive mathematician. My father's advisor, Yakov Sinai, is equally impressive and arguably one of the founders of Dynamical Systems, a very interesting subfield of mathematics (and, in some ways, probability theory) in its own right.
Re: The Man Who Invented Modern Probability
#4> Kolmogorov armed a group of researchers with electromechanical calculators and charged them with the task of calculating the rhythmical structures of Russian poetry
And this is what "big data" and NLP were like before computers :).
One can only imagine what he could have done with a modern computer. (Which is particular hard to imagine because he's responsible for advances which ultimately led to these same modern computers...)
Also, as an aside, I'm very impressed by the quality of nautil.us articles. I've come a across a few randomly, and they've been uniformly high in quality, interesting and insightful.
Moreover, unlike much popular science/math reporting, the articles are not ridden with obvious errors and yet are easily followed by somebody not in the field. An impressive combination. (Not surprising for this particular article, I suppose: it was written by a professor teaching the history of mathematics at MIT.)
Re: The Man Who Invented Modern Probability
#5The treatment of the Paradox of the Great Circle in the article is not very good. If you're left wondering, Wikipedia helps:
http://en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_parado...
Re: The Man Who Invented Modern Probability
#6>For example, irrational numbers—those that cannot be written as fractions— almost surely have no pattern in the numbers that appear after the decimal point. Therefore, most irrational numbers are complex objects, because they can be reproduced only by writing out the actual sequence.
That's false. Pi is irrational, but there's many short algorithms to calculate it's digits.
Re: The Man Who Invented Modern Probability
#7>For example, irrational numbers—those that cannot be written as fractions— almost surely have no pattern in the numbers that appear after the decimal point. Therefore, most irrational numbers are complex objects, because they can be reproduced only by writing out the actual sequence. That's false. Pi is irrational, but there's many short algorithms to calculate it's digits.
It reads to be like the write heard someone say almost all and didn't realize that that has special mathematical meaning:
Re: The Man Who Invented Modern Probability
#8>For example, irrational numbers—those that cannot be written as fractions— almost surely have no pattern in the numbers that appear after the decimal point. Therefore, most irrational numbers are complex objects, because they can be reproduced only by writing out the actual sequence. That's false. Pi is irrational, but there's many short algorithms to calculate it's digits.
The key phrases there are "almost surely" and "most irrational numbers." There are irrational numbers like pi which can be defined algorithmically, but taken together as a set they have zero Lebesgue measure. (In fact, they're countable.)
Re: The Man Who Invented Modern Probability
#9> Kolmogorov armed a group of researchers with electromechanical calculators and charged them with the task of calculating the rhythmical structures of Russian poetry And this is what "big data" and NLP were like before computers :). One can only imagine what he could have done with a modern computer. (Which is particular hard to imagine because he's responsible for advances which ultimately led to these same modern…
>And this is what "big data" and NLP were like before computers :)
aside, I recently interviewed a candidate for a "Big Data" position. He got quite chatty about Kolmogorov & Measure Theory, so I quickly cooked up an interesting problem to relax his nerves. I opened the interview with "If the side of a cube comes from a uniform distribution between 1 and 5 inches, what's the expected volume of the cube ?" I thought he'll give me the answer in 10 seconds & we'll move on but instead he was completely stumped & just stared blankly. I asked him to use the whiteboard but he simply drew a cube and wrote a^3, a~[1,5], and then went back to staring at it. Finally after 10 minutes of pin-drop silence, my partner, a non-math guy, took over the interview & asked him some Hadoop related technology questions & he started speaking again.
Re: The Man Who Invented Modern Probability
#10The treatment of the Paradox of the Great Circle in the article is not very good. If you're left wondering, Wikipedia helps: http://en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_parado...
tl;dr Because Great Circles have measure 0, any treatment of probability on them must instead use limits that asymptotically approach Great Circles. Also because they have measure 0, such limits don't converge in a consistent fashion. As it turns out, the two most common limit constructions give different results for that particular problem.
The closest analogy I can think of is trying to assign a value to the expression 0/0. It doesn't have a value, but you can construct limits that appear to converge to it. One such limit is lim{x/x} as x approaches 0, which tends to 1. Another limit is lim{x log(x)} as x approaches 0, which tends towards 0. Both of these constructions have claim to being the value of 0/0, and they don't reconcile with each other.