Bayes' Theorem
Definition
The mathematical rule that relates prior odds, the likelihood ratio, and posterior odds. In the odds form: O(Hp|E) = O(Hp) x LR. This is the formal basis for Bayesian forensic inference and is equivalent to the probability form of the theorem but easier to apply in sequential evidence evaluation.
- Odds form
- O(Hp|E) = O(Hp) x LR
- Three components
- Prior odds, likelihood ratio, posterior odds
- Use
- Formal basis for Bayesian forensic evidence evaluation
- Advantage
- Easier to apply sequentially than the probability form
Common questions
Why do forensic statisticians prefer the odds form of Bayes' theorem over the probability form?+
The odds form lets each new piece of evidence multiply directly onto the running odds via its own likelihood ratio, so evidence can be evaluated one item at a time and combined by simple multiplication, which is more tractable in casework than repeatedly recomputing conditional probabilities.
What is the most common misuse of Bayes' theorem in court testimony?+
The prosecutor's fallacy, where the likelihood ratio or a match probability is presented as if it were the probability of innocence, conflating P(evidence given innocence) with P(innocence given evidence). These are only equal when the prior odds happen to be 1, which they rarely are.
Does Bayes' theorem require the expert to know the prior odds?+
No. The theorem still works if the expert reports only the likelihood ratio, leaving the fact-finder to combine it with their own assessment of the prior odds from the rest of the case.
Related terms
- Evaluative Reporting
- A framework for expressing forensic conclusions in terms of the probability of the evidence given the prosecution hypothesis versus the defence hypothesis...
- 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...
- 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...
- Posterior Odds
- The ratio of the probability of the prosecution hypothesis to the probability of the defence hypothesis after the forensic evidence is taken...
- Prior Odds
- The ratio of the probability of the prosecution hypothesis to the probability of the defence hypothesis before the forensic evidence in question...
- Propositions (Hp / Hd)
- The two competing hypotheses in evaluative reporting. Hp is the prosecution proposition (for example, the accused is the source of the trace),...
- Transposition Fallacy
- The error of treating the probability of the evidence given a hypothesis as though it were the probability of the hypothesis given...
- Validation Study
- An empirical study that tests the accuracy, precision, and error rate of a forensic method under realistic conditions. Validation data are required...
Explained in these topics
- How Numbers Enter Forensic ConclusionsA mathematical rule relating prior probability, the likelihood of evidence under competing hypotheses, and posterior probability. In odds form: posterior odds...
- Prior and Posterior Odds in Court
- The Role of Statistics in Evidence EvaluationA rule relating prior probability, the likelihood ratio, and posterior probability. In forensic terms: posterior odds = prior odds multiplied by the LR. The th...