Probability and Randomness via Twitter

 Oh, just another interesting Twitter thread:

https://twitter.com/svpino/status/1437793433255620633

Foundations, Randomness, Free Will, the Aaronson Oracle

To end the week, another wonderful new episode from PBS's Infinite Series, this time on the foundations of mathematics:

Also, sort of cool… in the commentary after the episode the show host, Kelsey Houston-Edwards, briefly mentions the Aaronson Oracle, which I was unfamiliar with, and which interactively demonstrates the difficulty of 'randomness.' It's a program from Scott that predicts a choice (generally succeeding well-over half the time, with two possible choices) that you will make in attempting to randomly press two computer keys:

Read a little about it here:

https://github.com/elsehow/aaronson-oracle

...and then try it out here:

http://people.ischool.berkeley.edu/~nick/aaronson-oracle/

Of Course Laplace Predicted Derek Would Do This

The irrepressible Veritasium (Derek Muller) offers a quick lesson on randomness, information, entropy, quantum mechanics, and chaos (in 10 minutes no less!):



One good lay-friendly book that covers/expands on some of this same ground is Charles Seife's "Decoding the Universe."

Just A Random Post Today

Fantastic, and mostly very comprehensible, article (long read) on randomness by Scott Aaronson over at American Scientist:

http://www.americanscientist.org/issues/pub/2014/3/the-quest-for-randomness/3

"Randomness" is a topic that potentially cuts across so many different boundaries… mathematics, logic, computer science, philosophy, psychology, physics, engineering… even if you think you're not particularly interested in it, you should be! The article touches on a number of topics, including Kolmogorov complexity, the Berry Paradox, quantum computing, random number generation, and best of all it's just the first of two pieces Aaronson is doing for Am. Scientist.

And over at MathTango right now another Friday list of links to catch up on for the weekend.

Meditating... on Chaos

Food for thought… an interesting passage from chapter 85 (on chaos theory, also known as "bifurcation theory") of Richard Elwes' "Math In 100 Key Breakthroughs" (pgs. 345-7):

"How can one produce a random number? In the late 1940's, John von Neumann proposed a very strange answer to that question.  He suggested that applying a simple algebraic rule a few times should do the job. The rule is to begin with some number, call it x, and then multiply x by (1 - x), and multiply the result by 4. That is to say: x --> 4 X x X (1 - x).
"There does not seem to be anything especially 'random' about this bit of algebra. Once the initial number is chosen, say x = 0.1, the result of applying the rule is then completely predetermined. But a little experimentation reveals von Neumann's insight. The sequence produced by this rule runs: 0.1, 0.36, 0.9216, 0.2890, 0.8219, 0.5854, 0.9708, and so on (each number given to 4 decimal places). There does not seem to be much of a pattern here, and in fact that is no illusion. You can extend the sequence for as long as you like and in fact no pattern will emerge. Someone who did not know the rule being used would find it virtually impossible to distinguish between this sequence and one produced by a genuinely random physical process such as radioactive decay."…

"Today, von Neumann's rule is known as the logistic map, and it is one of the simplest examples of mathematical chaos, a phenomenon which has been recognized in many different situations…."

"In von Neumann's pseudorandom number generator, everything rests on the number 4, known as the parameter. Changing that value completely alters the behavior of the system. If one replaces 4 with a new parameter of 2, the logistic map ceases to be chaotic. Instead, for any starting value, the sequence will quickly home in on a fixed value of 0.5. This is known as an attracting point of the system.
"Increase the parameter from 2 to 3.4, and something new occurs. After a while, the sequence will endlessly flicker back and forth between two values around 0.84 and 0.45. This is known as an attracting 2-cycle. Raise the parameter a little higher to 3.5, and this is replaced with an attracting 4-cycle, and then at 3.55, an attracting 8-cycle, and so on. As the parameter increases, the length of the attracting cycle keeps doubling 15, 32, 64, and so on. This behavior is what chaos theorists call a sequence of bifurcations."

He goes on to explain that the bifurcations end once the parameter hits a certain threshold value known as the Feigenbaum point (named after chaos theorist Mitchell Feigenbaum). Beyond that point (like "4" in the example) the produced sequence will act chaotically forever, producing the famous "butterfly effect" whereby two sequences beginning at only slightly different starting values "end up entirely unrecognizable from each other."

A Little Bit On Randomness

h/t to Dan Meyer for this one:

Random Samples and 'WEIRD' People

35+ years ago in college it troubled me that among the first lessons learned in basic statistics was the necessity for employing "random samples" in research studies, even while in reality samples I saw in academic literature were virtually never truly random (indeed, the very concept of a really "random sample" is questionable).
35 years later, nothing has changed, but a journal paper by some behavioral psychologists, now making the rounds, entitled "The Weirdest People in the World," is highlighting this concern over how representative research conclusions from limited samples can be for humans in general.

abstract here:

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1601785

fuller citation here:

Henrich, J., Heine, S. & Norenzayan, A. (2010) "The Weirdest People in the World?" Behavioral and Brain Sciences. [PDF] [Audio File Part I] [Audio File Part II] [Coverage in Science]

Neuroanthropology blog has addressed the paper in an extensive blog post:

http://tinyurl.com/2bsazch

"WEIRD" people, BTW, are those from "Western, Educated, Industrialized, Rich and Democratic societies."

I'm just glad to see the issue getting the attention it deserves (and I'd add that while the criticisms raised are being principally pointed at behavioral and psychology studies, even biological/physiological research in humans is not entirely immune from such questions/concerns).

Randomness (Is there such a thing?)

NPR report on 'randomness' (includes podcast):

http://www.npr.org/templates/story/story.php?storyId=125993584

Older Gregory Chaitin article on randomness here:

http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer.html