Transposition of the Conditional
Definition
The mathematical name for the error at the core of the prosecutor's fallacy. P(A | B) and P(B | A) are generally not equal; their relationship is given by Bayes' theorem. In court contexts, A is often 'innocence' and B is 'the observed evidence'.
- Field
- Probability and legal statistics
- Formal statement
- P(A|B) is not generally equal to P(B|A)
- Governing relation
- Bayes' theorem links the two via the priors
- Legal manifestation
- Prosecutor's fallacy and defence attorney's fallacy
- Legal history
- Named and analysed in statistics-and-law literature since the 1980s-90s
Common questions
What is the difference between this and the prosecutor's fallacy specifically?+
Transposition of the conditional is the general mathematical error; the prosecutor's fallacy is its specific courtroom instance, where the probability of the evidence given innocence gets presented as if it were the probability of innocence given the evidence.
Can transposition of the conditional favour the defence instead of the prosecution?+
Yes. When a low match probability is misread as a low probability of guilt without accounting for how many other people could also match, or when a rare feature's rarity is used to imply innocence outright, the same reversal error runs the other way.
How is the error corrected in court?+
Bayes' theorem is applied explicitly, or a likelihood ratio is presented instead of a bare match probability, so the fact-finder combines the evidence's strength with prior odds from the rest of the case rather than treating the evidence as decisive on its own.
Related terms
- Defence Fallacy
- The converse error of inflating the importance of the RMP by arguing that, because many people in the population share the profile,...
- Likelihood Ratio (LR)
- The ratio of two conditional probabilities: the probability of the observed evidence given the prosecution's hypothesis (same source), divided by the probability...
- Prosecutor's Fallacy
- The error of treating the RMP (or its reciprocal) as the probability that the defendant is innocent, or as the probability that...
- Prior Probability
- The probability of guilt (or innocence) based on all evidence other than the specific forensic match under consideration. Bayes' theorem requires a...
- Product Rule (Probability)
- The rule that P(A and B) = P(A) x P(B) holds only when A and B are independent. Applying it to correlated...
- Random Match Probability (RMP)
- The probability that a randomly chosen unrelated person from the relevant population would match the evidence profile by chance. A very small...
- Reference Class
- The population against which a probability or frequency is calculated. Choosing the wrong reference class, for example using general population allele frequencies...
Explained in these topics
- History of Statistical Evidence in CourtsThe logical error of swapping a conditional probability with its converse: claiming that P(E|H) = P(H|E). This is formally equivalent to the prosecutor's falla...
- The Prosecutor's Fallacy