information · rates
Bayes' theorem
A test says this person is ill. How often is that person really ill?
A test says this person is ill. Of the people it calls ill, how many are really ill?
The mix-up. Two questions sound alike. Of the people who are ill, how many does the test catch? And of the people the test calls ill, how many are really ill?
The people. With 10 of 100 ill, the test catches 9 ill people and also calls 6 well people ill.
The 90%. 9 of 10 is 90%, and that answers how many ill people the test catches, not how many called-ill people are really ill.
The calculation. 9 ill and 6 well are 15 people the test calls ill, and 9 of those 15 is 60%.
When 40 of the 100 are ill. 36 of the 40 called ill are really ill, which is 90%, because fewer well people were left to flag.
A rare note. Shannon information is how much that news surprises you. Most days are sunny, so a note that says sun tells you what you already thought. Rain is the unusual day, and that message carries the surprise. A sure message carries none.
A red shoe. Hempel's ravens, also called the confirmation paradox, asks what should count. "All ravens are black" can be read as "a thing that is not black is not a raven." A red shoe fits that reading. It is a very small piece of evidence about birds.
Green so far. Grue is a second word that fits the same emeralds. Green means green at any time. Grue means green before a future date, and blue after that date. Every emerald seen so far fits both words. They disagree about the next one.