The Typical Set

(Part of Math and Wellness Month.)

Say you have a biased coin that comes up Heads 80% of the time. (I like to imagine that the Heads side has a portrait of Bernoulli.) Flip it 100 times. The naïve way to report the outcome—just report the sequences of Headses and Tailses—costs 100 bits. But maybe you don't have 100 bits. What to do?

One thing to notice is that because it was a biased coin, some bit sequences are vastly more probable than others: "all Tails" has probability 0.2100 ≈ 1.268 · 10−70, whereas "all Heads" has probability 0.8100 ≈ 2.037 · 10−10, differing by a factor of sixty orders of magnitude!!

Even though "all Heads" is the uniquely most probable sequence, you'd still be pretty surprised to see it—there's only one such possible outcome, and it only happens a 2.037 · 10−10th of the time. You probably expect to get a sequence with about twenty Tails in it, and there are lots of those, even though each individual one is less probable than "all Heads."

Call the number of times we flip our Bernoulli coin N, and call the entropy of the coinflip H. (For the 80/20 biased coin, H is ⅕ lg 5 + 4/5 lg 5/4 ≈ 0.7219.)

It turns out for sufficiently large N (I know, one of those theorems, right?), almost all of the probability mass is going to live in a subset of 2NH outcomes, each of which have a probability close to 2−NH (and you'll notice that 2NH · 2−NH = 1).

May Is Math and Wellness Month

(Previously, previously.)

Do you ever spend five months in constant emotional pain waging a desperate and ultimately unsuccessful behind-the-scenes email campaign with the aim of securing a public clarification of a trivial philosophy-of-language issue because you're terrified that your robot cult's inability to correct politically-motivated philosophy errors implies that you've lost the Mandate of Heaven and are therefore unfit to prevent the coming robot apocalypse?

Yeah, me neither.

Did you know that May is Math and Wellness Month (source: me)?? Math and Wellness month is traditionally celebrated by performing super-well at one's dayjob, going to the gym a lot, and studying math in the evenings!

Concerning Frame Control Via Salient Scenarios

"We need to institutionalize people in order to prevent them from hurting themselves" has the same memetic-superweapon structure as "We need to torture terrorists to get them to tell us where they've hidden the suitcase nuke." The scenario as stated obviously has consequentialist merit (death is worse than prison, megadeaths are worse than torture), so you'd have to be some kind of huge asshole—or a former suspected terrorist—to say, "I claim that this hypothetical scenario is not realized nearly as often as you seem to be implying and therefore falsifiably predict that many of your alleged real-world examples will fall apart on further examination."

Object vs. Meta Golden Rule

"I know it might seem like a lot to ask, but I wouldn't hesitate to do the same for you if our positions were reversed."

"I don't doubt that. But I can't help but notice that it would be easier for you to say it if the fact that they aren't reversed is—somehow—not a coincidence."

April Is Separability Month

It is now April! Did you know that April is one of the months in which every compact metric space is separable?

Proof. Let it be April, and let M be a compact metric space. Because M is compact, it is totally bounded, so for all n∈ℕ, we can cover M with finitely many open balls of radius 1/n. The centers of all such balls are a countable set which we can call C. But C is dense, because an arbitrary point p∈M is a limit point of C: an ε-neighborhood of p must contain the center of one the balls in our covering of M with ε/2-balls. Thus M contains a countable dense subset.

Binge-Purge

$ history | grep freeciv
  605  freeciv
  606  sudo apt-get install freeciv
  607  sudo apt-get remove freeciv
  652  rm -rf ~/.freeciv/
  706  sudo apt-get install freeciv
  722  sudo apt-get remove freeciv
  735  rm -rf ~/.freeciv/
  752  sudo apt-get install freeciv
  754  sudo apt-get remove freeciv
  768  rm -rf ~/.freeciv/
  785  history | grep freeciv

"Give Anything"

As a freshman on my high school's cross country team, our captain told me that to be a good runner, you needed to love pain.

I objected: a great runner could love to race, I said, and endure the pain only for the sake of competing and winning.

It's only fifteen years later (practically one foot in the grave), that I now see that I was wrong and he was right.

You can run out of habit or you can run because Coach would notice if you skip practice, but you cannot run because of the strictly instrumental effect that not-running would have on your goals. Our minds aren't built that way; what is separable conceptually is not separable architecturally.

Ultimately, to not sacrifice the gift, you have to love pain. You have to love life.