Probative Value and the Weight of Evidence

Before any question of weight arises, evidence must clear the relevance hurdle. Relevance is a minimal test: does this evidence tend to make any contested fact more or less probable? The word 'contested' matters. If the defence does not dispute that a death occurred, evidence proving the death adds nothing and can be omitted. If the dispute is about identity, evidence bearing on identity is relevant; evidence about the deceased's childhood is not.

Forensic evidence almost always clears the relevance hurdle for the simple reason that it is produced by an examination of physical material from the case. A DNA profile from a sample taken from the crime scene is relevant by construction: it exists because of the case. What can make forensic evidence irrelevant is a mismatch between what it can prove and what is actually disputed. If a defendant admits handling the victim's phone and the question is whether the handling was lawful, a fingerprint match on the phone adds nothing to what is already conceded.

Once relevance is established, courts in most legal systems apply a balancing test: does the evidence's capacity to prove a fact (probative value) outweigh the risk that it will distort the fact-finder's reasoning (prejudicial effect)? The balance is not symmetric. Relevant evidence with high probative value and low prejudice is admitted freely. Evidence with moderate probative value and high prejudice may be excluded even though it is technically relevant.

The admissibility balance is a judicial decision, not a scientific one. The forensic expert's role is to help the judge understand the probative value side of the equation: how strong is the evidence scientifically, what is its error rate, what are the alternative explanations. The prejudice side is primarily a legal assessment. An expert who overstates the scientific certainty of their finding increases the prejudicial effect without increasing the genuine probative value, which is one of the most common misuses of forensic evidence.

The likelihood ratio (LR) is the formal statistical expression of evidential weight. It asks: if I observe this evidence, is it more consistent with the prosecution's story or the defence's story, and by how much? The LR is defined as the probability of observing the evidence if the prosecution hypothesis is correct, divided by the probability of observing the same evidence if the defence hypothesis is correct.

For a DNA profile comparison in a typical case, the prosecution hypothesis is that the evidence came from the suspect (H1) and the defence hypothesis is that it came from a random unrelated individual (H2). If the DNA profile probability for a random individual is one in ten million, the LR is approximately ten million: the evidence is ten million times more likely to be seen if H1 is correct than if H2 is correct. This does not mean the probability of guilt is ten million to one; the jury must combine this LR with all other evidence and reasoning about the prior probability.

1. Step 1: Define the competing hypotheses
   State H1 (prosecution) and H2 (defence) clearly before calculating. The choice of hypotheses changes the LR. H2 is not always 'a random person from the general population'. It might be 'an unidentified relative of the victim' or 'a regular visitor to the premises', depending on what the defence actually argues.
2. Step 2: Estimate the probability of the evidence under H1
   If the suspect is the source, what is the probability of observing this result? For a DNA match with no laboratory error, this is effectively 1 (the suspect's DNA matches their own). For a glass refractive index match with measurement variability, it requires knowing the method's precision.
3. Step 3: Estimate the probability of the evidence under H2
   What is the probability of a random unrelated person (or whatever H2 specifies) producing the same result? For DNA this comes from allele frequency databases. For other evidence types it may require a survey of reference populations or historical case data.
4. Step 4: Compute and interpret the ratio
   LR = P(E|H1) / P(E|H2). An LR of 1 is neutral. Above 1 supports H1; the larger the LR the stronger the support. Below 1 supports H2. A verbal scale is often used in court alongside the number.

The likelihood ratio tells you how much the forensic evidence should shift your belief, but it does not tell you where to start. Bayes' theorem formalises the combination. In the odds form, the posterior odds of the prosecution hypothesis equal the prior odds multiplied by the likelihood ratio.

Posterior odds = Prior odds × Likelihood ratio

If a person is identified as a suspect purely because their DNA appeared in a database search of millions of profiles, the prior odds of their being the offender (before considering the DNA match) are very low. Multiplying those low prior odds by a large LR may still produce a posterior probability of guilt that is less than overwhelming. This is one reason why a cold-hit DNA match (a match produced by searching a database rather than by testing a known suspect) should be interpreted with more caution than a match where the person was already a suspect from non-DNA evidence.

The prior probability is the jury's domain, not the expert's. The expert supplies the LR: the multiplication factor the new evidence contributes. The jury combines it with everything else (eyewitness accounts, alibi, motive, other physical evidence) to reach a posterior. An expert who presents a posterior probability of guilt has trespassed into the jury's function by smuggling in a prior the jury never agreed to.

Courts regularly encounter two complementary probability errors. Understanding both is important for forensic scientists who must present findings, for lawyers who must challenge or support them, and for anyone interpreting forensic conclusions.

• Prosecutor's fallacy: treating P(evidence | innocent) as if it were P(innocent | evidence). A DNA match probability of one in ten million does not mean there is a one-in-ten-million chance of innocence. It means, conditional on the defendant being the source, the observation is certain; conditional on a random person being the source, the observation has probability one-in-ten-million. Converting this to a probability of innocence requires a prior, which is the jury's job.
• Defence fallacy (the 'island problem' version): arguing that because, say, 2,000 people in a city of two million match the DNA profile, the evidence proves nothing about this defendant. This is wrong because it ignores the prior probability that the defendant was at the scene. If the defendant is already placed at the scene by independent evidence, the 2,000-person pool shrinks dramatically. The LR still needs to be combined with the prior, not treated as if the prior were zero.

The cases of Barry George (UK, 2008) and Lucia de Berk (Netherlands, originally convicted 2003, acquitted 2010) are two well-known instances where probability reasoning errors by prosecution experts contributed to wrongful convictions that were later overturned. In the George case, a single gunshot residue particle was treated as more probative than the science supported. In the de Berk case, the statistical probability of the pattern of deaths on her shifts was grossly miscalculated by an expert without the proper statistical training.

A juror presented with a likelihood ratio of 50,000 has a number. What they need is a meaning. Forensic scientists who testify have a duty not just to report the number correctly but to give the jury the conceptual tools to use it. Several practical approaches have emerged from research on expert communication and jury comprehension.

• Verbal scale alongside the number: calibrated verbal scales (weak, moderate, strong, very strong, extremely strong support) give the jury an intuitive anchor. These are not a substitute for the number but a supplement that prevents the jury treating every LR above 1 as equivalent.
• Plain-language explanation: something like: 'This evidence is 50,000 times more likely to be observed if the defendant is the source than if a random unrelated person is the source.' This is accurate, non-technical, and does not confuse the evidence probability with a probability of guilt.
• Stating the hypotheses explicitly: the jury should understand what H1 and H2 are before the LR is given. If the defence hypothesis is not 'a random person' but 'a close relative of the suspect', the LR will be much lower for DNA, and the jury needs to know this is being taken into account.
• Disclosing error rates: even a technically perfect analysis has a laboratory error rate. If the false-positive rate for a DNA analysis is one in 10,000, that is a cap on the evidential weight. No LR calculation based on population genetics can exceed what the practical false-positive rate allows, and the jury should be told what that rate is.