In traditional color theory, there is the color wheel. The color wheel is nice and symmetric. But have you ever wondered why yellow seems brighter than the other colors? Why, when the color wheel is converted to gray-scale, isn’t it a uniform gray?
Two RGB color wheels are shown, but the right one is converted to grayscale.
Today, the Laser Interferometry Gravitational-Wave Observatory (LIGO) reported the first observation of gravitational waves. You can read about it in The New York Times (warning: autoplay) or on Sean Carroll’s blog. (ETA: also see the explanation in comic form.) I went straight to Physical Review Letters.
As an undergrad, I did some work on LIGO. Specifically, I was a data analyst looking for exactly the kinds of gravitational waves here observed. Anyway, I’m happy to play the role of your local expert, providing some context and answering any questions.
Today I’ll look at the paper that established information theory: “A Mathematical Theory of Communication“, by Claude Shannon in 1948. (mirror link)
We’ll start out with my favorite sentence in the paper:
THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRITER THAT THE CHARACTER OF THIS POINT IS THEREFORE ANOTHER METHOD FOR THE LETTERS THAT THE TIME OF WHO EVER TOLD THE PROBLEM FOR AN UNEXPECTED.
The above sentence shows glimmers of sense, which is astounding given that it’s generated randomly without any considerations of grammar or meaning. Of course, this may not seem so impressive in a world where we have Google Autocomplete and the What would I say? Facebook app. But it all started here!
Shannon’s paper considers a very general problem: communication. Communication means taking a particular message and reproducing the message at a different location. Another way of thinking about it is that you are selecting the actual message from a large number of possible messages. Information is a quantitative measure of the ability to select the correct message
The Dunning-Kruger Effect states that people with the lowest competence tend to overrate their competence, but people with the highest competence tend to underrate themselves. This was shown in 1999 paper by Dunning and Kruger which won an Ig Nobel Prize. Here’s one of the figures from the paper:
This figure shows scores from a humor test (where a “high” score means good agreement with professional comedians). There are similar figures for tests on grammar and logic.
The Dunning-Kruger effect has entered popular wisdom, and is frequently brought up when people feel like they’re dealing with someone too stupid to know how stupid they are.
But I have to admit that the popular wisdom led me wrong. I had a misconception: I thought that people’s self-assessment was actually anti-correlated with their competence. As you can see from the above figure, this is plainly not the original finding. People with lower competence tend to overrate themselves, but their self-assessments don’t quite surpass the self-assessments of people with higher competence.
On diminishing returns
It is intuitively true that the marginal utility of a dollar decreases as you have more money. If you earn $2000 dollars a month, the first $1000 prevents you from living on the streets, while the next $1000 pays for less essential needs. Because you can freely allocate your money, you will generally allocate it to the most important things first, and less important things second.
This is important in anything that involves chance, such as investment or gambling. For example, suppose you have $1000, and you have the opportunity to bet it all for a 50% chance of winning an additional $1000. You would prefer not to take that bet, because:
U($1000) > 0.5 U($2000) + 0.5 U($1000)
where U is the utility function. Even though the average payoff in dollars is zero; the average payoff in utility is negative. This leads to other nice results, including the notion that risk is bad, and the notion that giving money to poor people is good.
But what exactly is the functional form of U?
I occasionally read social science papers, and I think it’s worth sharing what I find, even though in principle you could read about it yourself. This time I read about statistics on Post-traumatic stress disorder (PTSD) in the US, and my particular interest is in PTSD from sexual assault and rape.