Smoke from the wildfires in Bastrop, Texas.
Photo by Kerri West. Full-sized image here. More Bastrop photos by Kerri West here.
At least two people have died in the fires and many have lost their homes.
I’ve taken my daughters a couple times to a father-daughter retreat at Wilderness Ridge Camp near Bastrop. That camp has been completely destroyed by the fires.
The “anti-calculus proposition” is a little result by Paul Erdős that contrasts functions of a real variable and functions of a complex variable.
A standard calculus result says the derivative of a function is zero where the function takes on its maximum. The anti-calculus proposition says that for analytic functions, the derivative cannot be zero at the maximum.
To be more precise, a differentiable real-valued function on a closed interval takes on its maximum where the derivative is zero or at one of the ends. It’s possible that the maximum occurs at one of the ends of the interval and the derivative is zero there.
The anti-calculus proposition says that the analogous situation cannot occur for functions of a complex variable. Suppose a function f is analytic on a closed disk and suppose that f is not constant. Then |f| must take on its maximum somewhere on the boundary by the maximum modulus theorem. Erdős’ anti-calculus proposition adds that at the point on the boundary where |f| takes on its maximum, the derivative cannot be zero.
“I use Emacs, which might be thought of as a thermonuclear word processor.” — Neal Stephenson
From In the beginning was the command line
If a study is completely infeasible using traditional statistical methods, Bayesian methods are probably not going to rescue it. Bayesian methods can’t squeeze blood out of a turnip.
The Bayesian approach to statistics has real advantages, but sometimes these advantages are oversold. Bayesian statistics is still statistics, not magic.
From Jaron Lanier:
Even in the places that are called anarchistic, in fact, what happens is a new kind of order, which is often very oppressive if you don’t happen to fit in. In San Francisco you can be attacked by mobs of bicycling advocates who’ve occasionally been quite ruthless because they believe in bicycles, and they think that they’re the most enlightened, free people in the world, and yet if somebody doesn’t agree with them, then they have trouble.
Similarly, Burning Man, which people who fit in at Burning Man must perceive is the most open, accepting place in the world is, in fact, extraordinarily unaccepting of people who don’t conform.
It’s one thing to love your neighbor in the abstract. It’s quite another to love your literal neighbor.
As G. K. Chesterton explains:
We make our friends; we make our enemies; but God makes our next-door neighbor. … The duty towards humanity may often take the form of some choice which is personal or even pleasurable. That duty may be a hobby … We may be made as to be particularly fond of lunatics or specially interested in leprosy … But we have to love our neighbor because he is there — a much more alarming reason for a much more serious operation. He is the sample of humanity actually given us.
Quote found in From the Library of C. S. Lewis
When asked why he robbed banks, Willie Sutton famously replied “Because that’s where the money is.”
If you read about data analysis in high dimensions, you might hear someone say they’re focused on a thin shell because that’s where the probability is. For a multivariate normal distribution in high dimensions, nearly all the probability mass is concentrated in a thin shell some distance away from the origin.
What does that mean? Why is it true? How thin is the shell and what is its radius?
It seems absurd to say the probability is concentrated in a shell. The multivariate normal density has its greatest value at the origin and quickly decays as you move out in any direction. So most of the probability must be near the origin, right? No, because mass equals density times volume. The density decays quickly as you move away from the origin, but volume increases quickly. The product of the two is greatest at some radius away from the origin. That’s the shell.
The volume of a sphere in d dimensions is proportional to rd, so volume increases very quickly if d is large. For example, if d = 100, how much of the volume of a unit sphere is between a distance of 0.99 and 1 from the origin? Since 1100 – 0.99100 = 0.634, this says 63.4% of the volume is in the outer shell of thickness 0.01.
Since volume of a sphere is proportional to rd, the volume of a shell of radius r and thickness Δr is roughly proportional to d rd-1 Δr. When you multiply that volume by the probability density exp( –r2 / 2 ) you get that the probability mass in the shell is proportional to
rd-1 exp( –r2 / 2 ) Δr.
This leads to a χ distribution with d degrees of freedom. (Not the better known χ2 distribution.) This distribution has mode √(d-1) and variance 1. For large d, the distribution is approximately normal. So a multivariate normal in d dimensions with d large has roughly 95% of its probability mass in a shell of radius √d with thickness 4, two standard deviations either side of √d. (I’m approximating anyway, so I approximated √(d-1) as √d to make the conclusion a little simpler.)
The graph below gives the probability density of shells as a function of radius in dimensions 10 and 100.
Related post: Volumes of Lp unit balls
