Wednesday, September 9, 2009

Frequentist thinking, or just bad math?

In Steven Pinker's excellent book "How the Mind Works", he describes how people are bad at probability assessments, but are much better at frequency assessments (pg 348). It almost comes out and says that the brain is frequentist and not Bayesian, and it certainly implies it. He outlines how badly people do on the classic rare disease problem: "frequency of a disease is 0.01%, you take a test that is 99.99% accurate (false positives at 0.01%), you test positive. What is your chance of having the disease". People, even educated people (even in the medical fields) get this one wrong a lot.

Pinker contrasts this with "Think of 10,000 people, so we expect 1 to be infected and 9,999 to be not infected. You take the test, and the 1 person infected will almost certainly test positive, and we expect 1 person out of the 9,999 to test positive as well. We know that you tested positive, so what is your chance of having the disease?"

He claims that people are much better at getting the answer right. In my view, this is less about being good at calculating frequencies, and more about being bad at math. The second way of describing the problem pretty much sets up and carries out all of the "difficult" math, and then rounds so that all you have are small integer values. People do much better with that. If you want an example, not in probability, you can read my paper on "Teaching Energy Balance using Round Numbers: A Quantitative Approach to the Greenhouse Effect and Global Warming", which was motivated by the Weight Watchers system.

In the weight watchers system, counting calories (215+340+...) is replaced by dividing by 50 and rounding (4+7+...). Same result, but small numbers are easier to work with.

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