md`## Prosecutor's fallacy Probability and statistics are useful, and give us a principled means of reasoning about the world, especially under conditions of uncertainty. I think statistical literacy is important for those reasons, but moreover, statistical illiteracy is dangerous. We've already seen that applying Bayesian reasoning to medical diagnosis reveals that it's much less likely that you have a rare disease _even when you test positive for it_ than you might think. One even more pernicious fallacy is known as the "prosecutor's fallacy" and involves mistaking conditional probabilities. The heart of the fallacy is a fundamental misunderstanding of conditional probabilities, which can be neatly encapsulated by the incorrect equality > ${tex`P(A|B) = P(B|A)`} Now that we're familiar with Bayes' theorem, we can instantly recognise this as false - there are missing factors of ${tex `P(A)`} and ${tex `P(B)`}. ### An example in court Imagine a criminal case, where there is some evidence against the defendent. Further, imagine that the evidence is extremely unlikely to occur if the defendent was innocent. As statisticians, we'd describe this with a conditional probability and say "the probability of the evidence, given innocence is low". We'll encode a low probability graphically as a small dot. > ${tex`P(\rm{evidence} | \rm{innocence}) \sim`} Thinking now as a member of the jury, is it likely that this person is guilty? It's extremely unlikely that the evidence would be found were they innocent, after all. We might claim that it's likely they are guilty, and that, considering the evidence, the probability of innocence is low. > ${tex`P(\rm{innocence} | \rm{evidence}) \sim`} Hopefully, when phrased in terms of conditional probabilities, you have spotted our error. We just equated two different conditional probabilities > ${tex`P(\rm{innocence} | \rm{evidence}) = P(\rm{evidence} | \rm{innocence})`} committing the prosecutor's fallacy. We know this is wrong, because Bayes' theorem tells us the correct way to relate conditional probabilities: > ${tex`P(\rm{innocence} | \rm{evidence}) = \frac{P(\rm{evidence} | \rm{innocence})P(\rm{innocence})}{P(\rm{evidence})}`} Now, this doesn't mean that the defendent is innocent. It means we need to properly consider the prior probabilities of innocence and evidence. In the most egregious case, the prior probability of the evidence is low regardless of innocence, and the prior probability of innocence is high. In this case, the low probability of the evidence in general cancels out the low probability of the evidence given innocence and we find that actually, the defendent is likely to be innocent, > ${tex`P(\rm{innocence} | \rm{evidence}) \sim`}${tex`\sim`} `