# Prosecutor's Fallacy / Transposed Conditionals The **Prosecutor's Fallacy** essentially makes the incorrect assumption that $P(A \mid B) = P(B \mid A)$, which is not actually true. This is easiest seen if we let: $A = \text{being a woman}$ $B = \text{being pregnant}$ Clearly we can see: $P(A \mid B) \neq P(B \mid A)$ One of these cases the probability is near $1$, the others is actually unlikely. In the case of prosecution, this generally takes the form: $P(\text{Innocent} \mid \text{Evidence}) = P(\text{Evidence} \mid \text{Innocent})$ In my post [here](https://www.nathanieldake.com/Machine_Learning/08-Bayesian_Machine_Learning-01-Bayesian-Inference.html#How-accurate-is-the-test?-$\rightarrow$-Application-of-Bayes-Theorem) this is outlined nicely, but essentially the issue is that we are not taking into account the base rate of evidence and innocence. ### Direction of Causality See the Book of Why for how this relates to forward and backward probabilities, as well as the direction of causality and asymmetry of information. This is essentially [Confusion of the inverse](https://en.wikipedia.org/wiki/Confusion_of_the_inverse). --- Date: 20220202 Links to: Tags: References: * [Part B - The Transposed Conditional (1): Prosecutor's Fallacy - Week 4 - Trials by Numbers or Numbers on Trial | Coursera](https://www.coursera.org/lecture/challenging-forensic-science/part-b-the-transposed-conditional-1-prosecutors-fallacy-lNh8f) * [The Prosecutor's Fallacy - YouTube](https://www.youtube.com/watch?v=V4cMWoGxEwo) * [Nathaniel Dake Blog](https://www.nathanieldake.com/Mathematics/03-Probability-02-Bayes-Rule.html) * [Nathaniel Dake Blog](https://www.nathanieldake.com/Machine_Learning/08-Bayesian_Machine_Learning-01-Bayesian-Inference.html#2.2.2-Inverse-and-Forward-Probabilities) * [Confusion of the inverse - Wikipedia](https://en.wikipedia.org/wiki/Confusion_of_the_inverse) * [Prosecutor's fallacy - Wikipedia](https://en.wikipedia.org/wiki/Prosecutor%27s_fallacy)