Prosecutor's Fallacy - Mathematical Analysis

Mathematical Analysis

Finding a person innocent or guilty can be viewed in mathematical terms as a form of binary classification. If E is the observed evidence, and I stands for "accused is innocent" then consider the conditional probabilities:

• P(E|I) is the probability that the "damning evidence" would be observed even when the accused is innocent (a "false positive").
• P(I|E) is the probability that the accused is innocent, despite the evidence E.

With forensic evidence, P(E|I) is tiny. The prosecutor wrongly concludes that P(I|E) is comparatively tiny. (The Lucia de Berk prosecution is accused of exactly this error, for example.) In fact, P(E|I) and P(I|E) are quite different; using Bayes' theorem:

Where:

• P(I) is the probability of innocence independent of the test result (i.e. from all other evidence) and
• P(E) is the prior probability that the evidence would be observed (regardless of innocence):

• P(E|~I) is the probability that the evidence would identify a guilty suspect (not give a false negative). This is usually close to 100%, slightly increasing the inference of innocence over a test without false negatives. That inequality is concisely expressed in terms of odds:

The prosecutor is claiming a negligible chance of innocence, given the evidence, implying Odds(I|E) -> P(I|E), or that:

A prosecutor conflating P(I|E) with P(E|I) makes a technical error whenever Odds(I) >> 1. This may be a harmless error if P(I|E) is still negligible, but it is especially misleading otherwise (mistaking low statistical significance for high confidence).