Causality—Bayes' Theorem

In this lesson, our objective is to get familiar with one of the most crucial formulas in the realm of probability: Bayes' theorem. This powerful theorem is fundamental to scientific advancements, serves as a key instrument in the field of artificial intelligence, and has even been employed in treasure hunting.

For instance, in the 1980s, Tommy Thompson used Bayesian search techniques to locate the shipwreck of the SS Central America from a century and a half prior, laden with gold now valued at approximately $700,000,000. You can read the full history in Ship of Gold in the Deep Blue Sea by Gary Kinder, which provides a detailed account of the treasure-hunting adventure.

Bayes' theorem is an essential concept that is well worth mastering.

Judgment under uncertainty

We explore the fascinating world of Bayes' theorem, starting with an intriguing example drawn from the study conducted by psychologists Daniel Kahneman and Amos TverskyTheir groundbreaking research on human judgment and decision-making was published in a series of articles, with the story of Steve featured in their paper titled "Judgment under Uncertainty: Heuristics and Biases Vol. 185, Issue 4157, pp. 1124-1131. " in the journal Science..Vol. 185, Issue 4157, pp. 1124-1131.

The story goes like this: Steve is a man who is very shy and withdrawn, with very little interest in people or the world of reality. He requires order and structure and a passion for detail. Now, consider this question: Which of the following descriptions do you find more likely, "Steve is a librarian" or "Steve is a farmer"?

Kahneman and Tversky's study found that most people, after hearing the description of Steve, were more likely to say he is a librarian than a farmer. This was because Steve's traits seemed to align better with the stereotypical view of a librarian than that of a farmer. However, the researchers pointed out that this judgment might be considered irrational, as people generally failed to incorporate the ratio of farmers to librarians into their judgments.

We can start by imagining a representative sample of farmers and librarians in a small city, for instance, 200 farmers and ten librarians. To make it easier to understand. Let's use the image below, where each square symbolizes a person. The ones with gray backgrounds are librarians and the ones with green backgrounds are farmers.