Base-rate fallacy, also called prosecutor’s Fallacy, is a cognitive bias where people ignore the base rate (general prevalence) in favor of event-specific information when making judgments or decisions.
Base-rate fallacy can be view the confusion of the posterior probability with the prior probability. When we specifically refer to this misunderstanding, it’s also known by several other terms such as transposed conditional fallacy, confusion of the inverse, conditional probability fallacy or the inverse fallacy.
An example of the base rate fallacy is the false positive paradox, where there are more false positive test results than genuinely positive result. For example, if a facial recognition camera can identify criminals 99% accurately, but analyzing 10000 people a day, then it will get 100 false positives. the high accuracy is outweighed by the number of tests, and the program’s list of criminals will likely have far more false positives than true. 1
Bayesian Approach
Note that according to Bayes’ theorem, the correct way to compute the posterior probability is
where we need to weigh the base rates
Examples
DNA Match
Suppose that there are 5 million people in Sydney. A murder occurs with DNA left on the weapon and a person matching the DNA is arrested.
Incorrect argument: given that the chance of a DNA match is 1 in 500000, the chance that the arrested person is guilty is very high.
We can compute the conditional probabilities of prior and posterior:
Note that
Sally Clark Case
Aside from multiplying the probability of two events as if they are independent, another error made by the persecutors of the Sally Clark Case is to confuse
If we use the Bayes’ theorem, then
This approach would have considered both the rarity of double SIDS and the rarity of double infant murder, leading to a very different conclusion. 2