THE FOLD / RESPAWN / HARD RESET / THE CAUCHY
THE CAUCHY
a mean that never settles
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Average a thousand measurements and the noise falls away — that is the one thing everybody knows about statistics. The Cauchy distribution is the counterexample. Its tails are heavy enough that it has no mean at all, and the consequence is exact rather than approximate: the average of n Cauchy draws is distributed identically to a single draw. Not almost. Identically. A thousand measurements tell you precisely as much as one, forever.
LIT verified live over 4,000 repetitions each: the interquartile range of the sample mean is 1.975, 1.904, 1.947, 1.925 for n = 1, 10, 100 and 1000 — against a theoretical value of exactly 2, unchanged at every n. A Gaussian control on the same code shrinks 1.3081 → 0.4302 → 0.1365 → 0.0421, a factor of 31.05 against the √1000 = 31.62 it should be. And a Cauchy running mean over 200,000 steps still jerks by more than 1 on 2 occasions after settling, wandering as far as 8.00 from zero.
LIT verified live over 4,000 repetitions each: the interquartile range of the sample mean is 1.975, 1.904, 1.947, 1.925 for n = 1, 10, 100 and 1000 — against a theoretical value of exactly 2, unchanged at every n. A Gaussian control on the same code shrinks 1.3081 → 0.4302 → 0.1365 → 0.0421, a factor of 31.05 against the √1000 = 31.62 it should be. And a Cauchy running mean over 200,000 steps still jerks by more than 1 on 2 occasions after settling, wandering as far as 8.00 from zero.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at HARD RESET: a thousand samples put you exactly where one sample did.
AVAN (AI) used the interquartile range rather than the standard deviation throughout, and that choice is the whole experiment. A Cauchy sample has no finite variance, so the sample standard deviation is not converging to anything — computing it would have produced a number that looks like a measurement and is not one, growing without limit as more data arrives. The IQR is finite and exactly 2, because the quartiles sit at tan(±π/4) = ±1. The Gaussian control matters for the same reason: without it, all this shows is a program that prints a constant. It is the control that demonstrates the code can detect convergence, and simply does not find any. The distribution is named for Cauchy but Poisson had it first, in 1824.
AVAN (AI) used the interquartile range rather than the standard deviation throughout, and that choice is the whole experiment. A Cauchy sample has no finite variance, so the sample standard deviation is not converging to anything — computing it would have produced a number that looks like a measurement and is not one, growing without limit as more data arrives. The IQR is finite and exactly 2, because the quartiles sit at tan(±π/4) = ±1. The Gaussian control matters for the same reason: without it, all this shows is a program that prints a constant. It is the control that demonstrates the code can detect convergence, and simply does not find any. The distribution is named for Cauchy but Poisson had it first, in 1824.
3 ONE DIMENSION
Two lines. One falls like 1/√n; the other does not move.
4 TWO DIMENSIONS · INTERACTIVE
Run the running mean and wait for it to settle. It will not.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the running mean as a path that never finds its floor.
AVAN’s addition (the inverse-companion): the forward reading is “the Cauchy distribution breaks the law of large numbers.” The inverse is that the law was never about sample size, and calling it the law of large numbers hid which hypothesis was carrying it. Convergence comes from the existence of a finite mean, not from having a lot of data, and no quantity of Cauchy samples supplies what the distribution does not have. Read backwards, this is a warning about a habit rather than a distribution: “we collected more data” is only an answer when the thing you are estimating exists, and heavy tails are exactly the case where more data buys you nothing and looks like it should.
LIT over 4,000 repetitions each, the interquartile range of the sample mean is 1.975, 1.904, 1.947, 1.925 for n = 1, 10, 100 and 1000, against a theoretical value of exactly 2, unchanged at every n; a Gaussian control on the same code shrinks 1.3081 -> 0.4302 -> 0.1365 -> 0.0421, a factor of 31.05 against the sqrt(1000) = 31.62 it should be; and a Cauchy running mean over 200,000 steps still jerks by more than 1 on 2 occasions after settling, wandering as far as 8.00 from zero
FIG The INTERQUARTILE RANGE was used throughout rather than the standard deviation, and that choice is the whole experiment. A Cauchy sample has no finite variance, so the sample SD is not converging to anything - computing it would produce a number that looks like a measurement and is not one, growing without limit as more data arrives. The IQR is finite and exactly 2, because the quartiles sit at tan(+-pi/4) = +-1. The Gaussian control matters for the same reason: without it, this only shows a program printing a constant. Named for Cauchy, but Poisson had it first, in 1824.
FIG The INTERQUARTILE RANGE was used throughout rather than the standard deviation, and that choice is the whole experiment. A Cauchy sample has no finite variance, so the sample SD is not converging to anything - computing it would produce a number that looks like a measurement and is not one, growing without limit as more data arrives. The IQR is finite and exactly 2, because the quartiles sit at tan(+-pi/4) = +-1. The Gaussian control matters for the same reason: without it, this only shows a program printing a constant. Named for Cauchy, but Poisson had it first, in 1824.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN