Prior and Posterior Odds in Court

Bayes' theorem in its odds form is: posterior odds = prior odds x likelihood ratio. Each term has a precise meaning and a clear owner.

The prior odds reflect everything the fact-finder knows before the scientist gives evidence. In a robbery case where a defendant was found near the scene carrying property matching the victim's description, the prior odds before the DNA evidence is heard might be relatively high. In a case resting entirely on a cold-case database hit with no other evidence, the prior might be lower. The scientist does not know which case they are in and should not try to guess.

The LR summarises what the scientist's examination adds. An LR of 1 means the evidence is equally probable under both propositions: the forensic examination has provided no discriminating power. An LR of 10 means the evidence is ten times more likely if Hp is true than if Hd is true. An LR of 0.1 means the evidence is ten times more likely under Hd: the scientist's evidence supports the defence proposition. The direction matters as much as the magnitude.

The prior odds are formed from everything that is known about the case before the forensic evidence is admitted. This includes eyewitness identification, surveillance footage, the defendant's location at the relevant time, the defendant's prior relationship to the scene or victim, and the background frequency of the offence in the relevant population. None of these are scientific measurements: they are factual and circumstantial materials that the fact-finder weighs according to legal rules of evidence.

In practice, neither a judge nor a juror assigns an explicit numerical value to the prior odds. They carry an implicit sense of how probable guilt seems before hearing the forensic evidence, and the scientist's LR updates that sense. This implicit approach is how Bayesian reasoning operates in real trials, even when the word 'Bayesian' is never used. The framework is nonetheless useful for understanding what the scientist's evidence can and cannot do: it can move the odds, but cannot set them.

A specific concern arises when the forensic evidence is the primary evidence against the defendant, such as in a cold-hit DNA case where the database match is the only connection between the defendant and the crime. Some statisticians have argued that the appropriate prior in such a case is 1 divided by the number of people in the database, making the prior odds very low. If the LR is large, the posterior odds may still support conviction, but the calculation makes clear that an LR of 1 in 10,000 from DNA evidence in a database of 10,000 people leaves the posterior odds at exactly 1:1, which is not enough for conviction.

See also: Random Match Probability for the related concept of how a single source probability is calculated without reference to prior odds.

The transposition fallacy is the error of treating P(E|Hd) as equivalent to P(Hd|E). In plain language: the probability of seeing this evidence if the defendant is innocent is not the same as the probability of the defendant being innocent given this evidence. The two quantities can differ by orders of magnitude depending on the prior odds.

The error appears in many forms. A forensic biologist who says 'the probability that someone other than the defendant is the source of this DNA is 1 in a million' has correctly described P(E|Hd) if the reference population is specified. If the biologist then says 'the probability that the defendant is innocent is 1 in a million', they have committed the transposition fallacy: that second statement is the posterior probability of innocence, which requires the prior odds the scientist does not have.

The Netherlands case of the Lucia de Berk prosecution (2003 to 2010) is an example of both the transposition fallacy and the base-rate problem operating together. The statistical expert calculated the probability of so many patient deaths occurring by chance on the shifts of one nurse as 1 in 342 million. The court treated this as the probability of innocence, which it is not. After a decade of litigation, her convictions were quashed. The case has since been studied in probability and law curricula in many countries as a demonstration of what goes wrong when the prior is not handled separately.

In the United States, the Daubert standard (Daubert v Merrell Dow Pharmaceuticals, 509 US 579, 1993) requires federal courts to screen expert testimony for scientific validity and relevance. Testimony that transposes probabilities may fail the reliability criterion. State courts apply varying standards but the same logical error is equally damaging. In India, expert testimony is governed by Section 45 of the Bharatiya Sakshya Adhiniyam 2023 (which replaced the Indian Evidence Act 1872), and while the statute does not specify probability formats, the same error of stating posterior probabilities as if they were LRs would be a logical defect that courts can weigh against the testimony.

The forensic scientist's task is to calculate and communicate the LR for the evidence they have examined, under clearly stated propositions. The propositions are usually set at source level (this mark came from the suspect's shoe vs this mark came from some other shoe) rather than offence level (the defendant committed the burglary vs the defendant did not). The scientist works at the level they have expertise in; moving to offence-level propositions requires access to all the case evidence, which the scientist typically does not have.

The European Network of Forensic Science Institutes (ENFSI) published a guideline on evaluative reporting in 2015, recommending the LR framework as the appropriate format for forensic conclusions. The UK Forensic Science Regulator's Codes of Practice require providers to use an evaluative framework consistent with Bayesian principles. The Association of Forensic Science Providers (AFSP) in the UK has issued similar guidance. These documents do not all require a numerical LR: verbal scales that map qualitative terms to LR ranges are permitted, provided the scale is defined and applied consistently.

A common verbal scale pairs terms with LR ranges: 'limited support' might correspond to LR 2 to 10, 'moderate support' to LR 10 to 100, 'strong support' to LR 100 to 1000, 'very strong support' to LR 1000 to 1,000,000, and 'extremely strong support' to LR above 1,000,000. The precise boundaries vary between providers and countries. The UK's Forensic Science Regulator has noted that different scales in use across providers are a source of inconsistency and has encouraged standardisation.

The allocation of the prior odds to the fact-finder is not merely a convention: it reflects the constitutional structure of criminal trials in most systems. In jury systems, the jury evaluates the whole of the evidence and reaches a verdict. No single piece of evidence, including forensic evidence, determines the verdict mechanically. The judge's role is to direct the jury on the legal standard of proof (beyond reasonable doubt in criminal cases) and to evaluate the admissibility of expert testimony, but not to substitute their own numerical prior for the jury's judgment.

In bench trial systems, where the judge is also the fact-finder, the same logic applies: the judge must distinguish between the LR supplied by the expert and their own prior assessment of the non-scientific evidence. The formal Bayesian framework helps to keep these separate, even if the judge does not use arithmetic explicitly. The alternative, where a judge or jury simply accepts the scientist's conclusion about the posterior, is a delegation of the fact-finding function that the legal system is designed to prevent.

Under the Bharatiya Nagarik Suraksha Sanhita 2023 (which replaced the CrPC in India), sessions courts and magistrate courts are the primary fact-finders in serious criminal cases. The principle that the court, not the expert, determines guilt is the same as in common law systems. The European Convention on Human Rights (Article 6, right to a fair trial) has been interpreted by the European Court of Human Rights to require that the fact-finder not simply adopt an expert conclusion without independent evaluation. The same principle appears in the US Constitution's Sixth Amendment right to jury trial.

For background on how statistical reasoning entered courts historically, see History of Statistical Evidence in Courts.

How the LR is communicated to a court is a live research and policy question. DNA evidence, where LRs of billions are common, is typically reported numerically: a random match probability of 1 in 10 billion is a direct statement of P(E|Hd) and is widely understood by fact-finders as very strong evidence. For disciplines where the LR is harder to calculate with precision, such as footwear comparison, fingerprint comparison, or handwriting analysis, a verbal scale is more common.

Jury research has shown that people have systematic difficulties with conditional probability statements. The confusion between P(E|Hd) and P(Hd|E) is not confined to lawyers and experts: jurors make the same error. Some researchers advocate presenting evidence as natural frequencies ('in a population of 10,000 people unrelated to the victim, about 1 would match this profile') rather than probabilities, arguing that frequency formats reduce the transposition error. Other researchers have found that numerical LRs of very large magnitude are treated by jurors as essentially equivalent to certainty, which may itself distort verdicts.

A specific issue arises with very low LRs from non-DNA evidence. If a footwear expert reports an LR of 10 (the mark is 10 times more likely to have come from the defendant's shoe than from a random shoe in the population), this is a meaningful evidential contribution, but jurors may regard it as weak and largely disregard it. Research on anchoring and numeracy suggests that the magnitude of the LR is not linearly perceived: the difference between LR = 10 and LR = 1000 looms larger than the difference between LR = 100,000 and LR = 10,000,000, even though both are differences of two orders of magnitude.

These communication challenges do not undermine the Bayesian framework: they are problems of presentation and comprehension that courts and scientists are still working through. They do underscore why the scientist's role is to provide the LR clearly and correctly, why the judge must direct the jury carefully on what an LR means and does not mean, and why the jury's independent evaluation of all evidence, including the prior information, remains essential. See also Numbers in Forensic Conclusions for more on the challenge of translating numerical evidence for non-specialist audiences.