The Likelihood Ratio: Definition and Logic

Let Hp be the prosecution hypothesis (for example, that the defendant is the source of the DNA found at the crime scene) and Hd be the defence hypothesis (for example, that an unknown person is the source). Let E be the observed evidence. The likelihood ratio is:

LR = P(E | Hp) / P(E | Hd)

P(E | Hp) is the probability of observing the evidence if Hp is true. For a DNA match where the defendant is the true source and the test has no false-positive rate, this probability is close to 1: if the defendant is the source, we expect a match. P(E | Hd) is the probability of observing the evidence if Hd is true: if an unknown person is the source, the probability of a match is the random match probability in the relevant population, typically a very small number.

The ratio LR = 1 / RMP when P(E | Hp) = 1, which is often a reasonable approximation for high-quality DNA profiles from a single contributor. If the RMP is 1 in a million, the LR is 1,000,000: the evidence is one million times more probable under Hp than under Hd. This is the evidential weight of the observation, not a probability of guilt.

Bayes' theorem can be written in several forms. The odds form is most useful in forensic applications because it makes the role of the LR explicit:

Posterior odds = Prior odds × LR

Odds are the ratio of the probability of an event to the probability of its complement. If a hypothesis has probability p, its odds are p / (1 - p). Odds of 1:1 (equal odds) correspond to a probability of 0.5. Odds of 9:1 correspond to a probability of 0.9.

To use the formula: suppose the court has assessed prior odds of Hp to Hd at 1:1 (equal prior odds, a neutral starting point for illustration). The forensic expert reports an LR of 10,000. The posterior odds are 1 × 10,000 = 10,000:1 in favour of Hp. Converting to probability: 10,000 / (10,000 + 1) is approximately 0.9999. A prior probability of 0.5 has become a posterior probability of approximately 0.9999 after this single piece of evidence.

Now change the prior. Suppose the case has a credible alibi and the prior odds are 1:99 against Hp (prior probability of Hp = 0.01). The posterior odds are (1/99) × 10,000 = 101:1 in favour of Hp. The posterior probability of Hp is approximately 0.99. The same LR produces a different posterior probability depending on the prior. This is why the expert reports the LR, not the posterior probability: the expert does not know the prior, and the prior is legitimately a matter for the court.

The prosecutor's fallacy is one of the most consequential errors in forensic evidence interpretation. It consists of treating P(E | Hd), the probability of the evidence under the defence hypothesis, as if it were P(Hd | E), the probability of the defence hypothesis given the evidence. These two quantities are not equal, and confusing them can transform a legitimately informative piece of evidence into a statement that sounds like near-certain guilt.

A concrete example: a DNA random match probability of 1 in 10 million. The prosecutor's fallacy renders this as: there is only a 1 in 10 million chance the defendant is innocent. This is wrong. The RMP is P(E | Hd): the probability of seeing a match if an unrelated person is the source. P(Hd | E) is the posterior probability of innocence given the match, which depends on the prior odds. If the prior probability of guilt is very low (the defendant was identified only because their profile was in a database of millions), the posterior probability of guilt after the match may itself be much less than 1.

The defence fallacy is the mirror error: treating P(E | Hd) as if evidence with a moderate RMP is therefore uninformative. An RMP of 1 in 1000 is not a small number in absolute terms, but an LR of 1000 is still strongly supportive of Hp relative to Hd. The LR framing correctly captures this by placing both probabilities on equal footing.

Before evaluative reporting frameworks became standard, forensic DNA reports typically stated a random match probability alone. The RMP is P(E | Hd): the probability that a randomly chosen person from the reference population would produce evidence consistent with the crime sample. It is a real and useful number, but it is incomplete as a measure of evidential weight because it gives no information about P(E | Hp).

The shift from reporting the RMP to reporting the LR is not merely cosmetic. It forces the expert to consider and state the probability of the evidence under both hypotheses, which requires a clear formulation of what each hypothesis actually says. This process often reveals ambiguities in the propositions that, if left unstated, would make the evidence uninterpretable. See Numbers in Forensic Conclusions for how numerical statements translate into court reports.

The proper division of roles is clear in the Bayesian framework. The forensic expert evaluates the evidence under the competing hypotheses and reports the LR or, where a precise numerical LR is not available, a verbal equivalent on a calibrated scale. The expert does not set the prior odds: the prior odds depend on all other evidence in the case, witness testimony, CCTV, opportunity, motive, and the plausibility of the defence account. These are matters for the court.

In many jurisdictions this division has now been codified or endorsed. In England and Wales, the guidance from the Crown Prosecution Service and the Forensic Science Regulator both draw on the ENFSI evaluative reporting framework. In the Netherlands, the Netherlands Forensic Institute has used LR-based reporting since the late 1990s. In Australia, forensic science standards developed by the National Institute of Forensic Science (NIFS, now part of the Australian and New Zealand Policing Advisory Agency) adopt the same approach. In India, court-appointed expert evidence is governed by the Bharatiya Sakshya Adhiniyam 2023 (replacing the Indian Evidence Act 1872), which preserves the principle that the expert gives an opinion on specialised knowledge and the court decides the ultimate issue: this is compatible with LR reporting. In the United States, the President's Council of Advisors on Science and Technology (PCAST) 2016 report on forensic science recommended that probabilistic reporting frameworks be adopted more broadly.

One practical consequence: a forensic expert who states in court that the evidence proves guilt, or that a defendant is almost certainly the source, has overstepped. The expert's role is to say how much more probable the evidence is under one hypothesis than the other. The ultimate question is reserved for the fact-finder, whether a judge, a jury, or a panel. This boundary is enforced by appellate courts in several jurisdictions and is a recurring subject of judicial guidance.

Many forensic evidence types do not yield a single precise numerical LR. Glass fragment comparisons, fibre evidence, tool marks, and handwriting analysis involve subjective assessments where probability estimation is approximate. In these cases, laboratories use verbal scales to convey the strength of the evidence, mapping ranges of numerical LR onto verbal descriptions.

The ENFSI scale, published in the 2015 Guideline for Evaluative Reporting, is the most widely adopted. It runs from statements supporting Hd (such as: the evidence is more probable under Hd, giving limited support to Hd) through neutral (LR close to 1, evidence does not distinguish the hypotheses) to statements supporting Hp at increasing strength (moderate support, strong support, very strong support, extremely strong support). A typical laboratory statement for a high-LR DNA result might read: the evidence is extremely strong support for the proposition that the defendant is the source of the DNA recovered from the item.

Numerical LR values are standard for forensic DNA, where population allele frequency databases and well-validated statistical models allow precise calculation. For other evidence types, the path from observation to LR estimate involves more subjective judgement, and the verbal scale both acknowledges that uncertainty and keeps the evidence within the proper inferential framework. Neither form, numerical nor verbal, gives the posterior probability of guilt: both require the court to combine the expert's contribution with the prior.