What does a 18th Century Philosopher have to Offer the 21st?
It is common to believe that our generation, has a monopoly of all the wisdom that is worth acquiring. But one of the great advances in social networking is the potential for rapid delivery of wisdom lost, from previous generations. Thomas Bayes was a 18th century philosopher who published small mathematical treatise on conditional probability. The practical import of his theorem was divined earlier by David Hume, who realized that when we are presented with testimonials that seem extraordinary we should focus on the possibility that the person testifying to this rare event is mistaken. That is, we should compare in our minds the chances that a "miracle" happened with the chances that the person was honestly mistaken about what they saw or reported.
In the late 1970's Amos Tversky and Daniel Kahneman rediscovered various interesting failures of individuals to use Bayes theorem - which they called the "Base Rate" problem, which might be thought of as the unrecorded chances of a miracle happening.. Gigerenzer & Hoffrage, in the mid 90's, challenged Tversky and Kahneman's assertion that individuals did not pay attention to the base rate, by recasting some their experiments and coming to different conclusions.
Over at the Science and Law Blog, the importance of Bayes theorem is stated this way.
"The importance of understanding base rates and Bayes' Theorem cannot be overstressed, particularly in the case of many types of medical and scientific testimony. The importance of base rates is seen in the following problem: A disease occurs in 1% of the population, and a test has been developed which has an 80% accuracy rate (i.e., if you have the disease, there is a 80% chance the test will pick it up), and a 9.6% false positive rate (i.e., if you don't have the disease, there is a 9.6% chance of getting a positive result anyway). Sam tests positive for the disease. What is the probability that Sam has the disease?The general inclination is perhaps to say 90.4%, because the false positive rate is 9.6%. This conclusion, however, is wrong because it does not account for the rarity of the disease in the general population. (As doctors are often trained to think, if you hear hoofbeats, think horses, not zebras.) Using Bayes' Theorem--and here I will spare the reader the mathematical details--one can show that the probability that Sam has the disease is 7.8%. Intuitively, this is because given the rarity of the disease, it is more likely that Sam is actually one of the false positives than one of the people with the disease. Short of being a math genius, however, crunching the numbers is extremely difficult to do intuitively, and merely plugging values into Bayes' formula has a certain mystical quality that might make jurors (or judges) skeptical.
Psychological research by Gigerenzer & Hoffrage, however, suggests that people find analyzing the problem from a frequentist perspective far easier than from the probabilistic perspective shown above. We can see this by transforming the example above to series of frequencies: A disease afflicts 10 out of 1000 people in the population. For people with the disease, 8 out of 10 will have positive test results. For people without the disease, the test will still (erroneously) yield a positive result 95 out of 990 times. Sam tests positive for the disease. What is the probability that Sam has the disease?
The answer follows far more simply. Out of a population of 1000 people, 8+95=103 people will test positive. And of these 103 people, only 8 actually have the disease, so the probability that Sam has the disease is 8/103 = 7.8%."
For the mathematically annoyed, it is helpful to look at the following chart.
| Test is Positive | Test is Negative | ||
| Person has Disease | 8 | 2 | 10 |
|
Persons has Not Disease |
95 | 895 | 990 |
| 103 | 897 | 1000 |
There are 1000 people, which explains the lower right corner. Since only 1% of the population gets the disease, then the last column must be 10 and 990. Finally, since the test is only 80% accurate, then the left to right diagonal must be 80% of last column. So if the test is positive, column 1, we have 103 individuals of which only 8 have the disease, 7.8% -which is more than 1%, but considerably lower than 80%.
The moral of the story is this: if a rare event (10/1000) is reported by a very reliable witness (80/100), the chances that the rare event happened is closer to its base rate (10/1000) than the accuracy of the reliable witness (80/100).
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