Yudkowsky presents Bayes' Theorem in an "excruciatingly gentle introduction." While many articles which explain Bayes' Theorem's application to one field or the other, it may be difficult to understand Bayes' Theorem at face value. As Yudkowsky writes, "It's this equation. That's all. Just one equation ... It looks like this random statistics thing." Compounding the problem is the fact that this mathematical concept is very counter-intuitive. It just seems to be one of those things that is inherently difficult for humans to grasp. This paper attempts to simplify Bayes' Theorem.
After introducing Bayes' Theorem with the standard mammography example, Yudkowsky introduces an even simpler example:
Suppose that a barrel contains many small plastic eggs. Some eggs are painted red and some are painted blue. 40% of the eggs in the bin contain pearls, and 60% contain nothing. 30% of eggs containing pearls are painted blue, and 10% of eggs containing nothing are painted blue. What is the probability that a blue egg contains a pearl?
He then sets up the problem using standard Bayesian notation:
- p(pearl) = 40%
- p(blue|pearl) = 30%
- p(blue|~pearl) = 10%
- p(pearl|blue) = ?
When set up in this manner, it is easier to plug the information into the formula to come to the correct answer, which is 66.7%. However, the most effective way to introduce Bayes' Theorem is using
Natural frequencies - saying that 40 out of 100 eggs contain pearls, 12 out of 40 eggs containing pearls are painted blue, and 6 out of 60 eggs containing nothing are painted blue. A natural frequencies presentation is one in which the information about the prior probability is included in presenting the conditional probabilities.
Yudkowsky mentions that presenting the problem in this way seems like cheating. However, in the real world, it is important to cheat, in the sense that the correct answer should be as obvious as possible.
The significance of Bayes' Theorem comes because it helps people make better sense of how new probabilities relate to one another. Because test cases are a measure of a sample, not a measure of the total population, it makes sense that forecasting accuracy increases with each test. So Bayes is a method to increase forecasting accuracy given new information. Given statistically independent evidence, calculations of probability will not change. But if the evidence is statistically linked, Bayes will allow the researcher to update the probability, regardless of the significance of new evidence.
In any case, the formula is still difficult to understand mathematically and cognitively.
Source: Yudkowski, Eliezer S. (n.d) An Intuitive Explanation of Bayes' Theorem. http://yudkowsky.net/rational/bayes