Bayes Theorem in Evidence Evaluation

Bayes theorem can be written in probability form or in odds form. The probability form is useful theoretically, but it requires the analyst to handle the full prior probability of each hypothesis, which demands knowledge of all case information. The odds form separates the equation into a part the scientist can supply and a part the scientist cannot.

The odds form is: Posterior Odds = Prior Odds x Likelihood Ratio. In notation, if Hp is the prosecution hypothesis and Hd is the defence hypothesis, and E is the evidence: P(Hp | E) / P(Hd | E) = [P(Hp) / P(Hd)] x [P(E | Hp) / P(E | Hd)]. The first bracketed term is the prior odds. The second bracketed term is the likelihood ratio.

The LR is a ratio, not a probability. It can range from zero to infinity. An LR of 1 means the evidence is equally consistent with both hypotheses and adds nothing. An LR of 1000 means the evidence is 1000 times more probable if Hp is true than if Hd is true. An LR of 0.001 means the evidence is 1000 times more probable under Hd, supporting the defence position. The LR is symmetric in this sense: evidence can support either side, and the direction is determined by which hypothesis makes the observed evidence more probable.

The LR is meaningless without a clear pair of competing hypotheses. In practice, the forensic scientist proposes hypotheses at the level appropriate to the findings, typically the source level (who contributed the material) or the activity level (what actions produced the trace). Conflating these levels is a common source of error.

A source-level pair for a glass transfer case might be: Hp, the glass fragments on the suspect's clothing came from the broken window at the scene; Hd, the glass fragments came from some other source. An activity-level pair might be: Hp, the suspect broke the window; Hd, the suspect was in the area but did not break the window. The LR values for these two pairs can differ substantially, and the relevant pair must match the question actually in dispute in the trial.

The choice of Hd requires care. In DNA cases, the realistic defence hypothesis is usually that the DNA came from a randomly selected individual from the relevant population. In a case with a defined suspect pool, a narrower Hd may be more appropriate. A poorly chosen Hd produces an LR that does not actually answer the question before the court, and courts in England, the Netherlands, and Australia have criticised evidence reports where the hypotheses were not clearly defined.

The prior odds represent the relative probability of Hp and Hd before the forensic evidence in question is weighed. Setting them requires incorporating every other piece of case information: witness accounts, the defendant's alibi, CCTV footage, motive evidence, and the legal presumption of innocence. A forensic scientist has no privileged access to most of this material.

The presumption of innocence, embedded in criminal procedure across most legal systems including under Article 6 of the European Convention on Human Rights, implies that, at the start of a trial, the prior odds should be set very low for Hp. But the exact numerical value is a legal and policy question, not a scientific one. Courts in England and Wales, drawing on the guidance in R v Adams [1996] 2 Cr App R 467, have held that it is not the scientist's role to assign numerical priors, precisely because doing so requires the scientist to act as a fact-finder.

The practical consequence is that the forensic scientist reports the LR and explains what it means. The fact-finder then applies the LR to whatever prior odds they have arrived at from all the other evidence. This division keeps the expert within the domain of their competence and preserves the fact-finder's role. It is the same principle applied under the Bharatiya Sakshya Adhiniyam 2023 in India, the Federal Rules of Evidence in the United States, and the Evidence Act 2006 in New Zealand: expert opinion assists the fact-finder but does not replace them.

Numerical LR values span many orders of magnitude. DNA LRs for a full 20-locus STR match in a non-relative case commonly exceed 10 billion. Glass refractive index LRs for a three-fragment transfer might be in the hundreds. Firearms toolmark LRs from some current models range from 10 to 10,000 depending on agreement quality. Communicating these values to a non-specialist jury requires both the number and a qualitative gloss.

The ENFSI verbal scale for the LR provides a consistent vocabulary across European forensic services. LR values between 1 and 10 are described as limited support for Hp. Values from 10 to 100 give moderate support. Values from 100 to 1000 give strong support. Values from 1000 to 10,000 give very strong support. Values above 10,000 give extremely strong support. Values below 1 are described in the corresponding terms supporting Hd. The scale is logarithmic in its boundaries, recognising that the perceptual difference between LR = 1,000 and LR = 1,100 is negligible, while the difference between LR = 10 and LR = 1,000 is substantial.

A related error is the defence fallacy: treating a high LR as proof of guilt rather than as evidence that shifts the odds. Both errors arise from treating the LR as if it were a posterior probability. The correct statement is always comparative: the evidence is X times more probable under Hp than under Hd.

Consider a DNA single-source profile recovered from a crime scene. The prosecution hypothesis is that the defendant is the contributor. The defence hypothesis is that an unknown unrelated individual from the same population is the contributor. The population frequency of the profile is 1 in 5 million. The LR is therefore 5 million: the evidence is 5 million times more probable if the defendant is the source than if a random person from the population is the source.

Now consider that the prior odds before this DNA evidence are 1:1000 (the police investigation placed the defendant at the scene but with some uncertainty, and the fact-finder has assessed this preliminary evidence as making Hp 1000 times less likely than Hd). Posterior odds = (1/1000) x 5,000,000 = 5,000. Posterior odds of 5,000:1 favour Hp strongly. These posterior odds translate to a posterior probability of Hp of approximately 5000/5001, or about 99.98%.

The same calculation structure applies to other evidence types. For glass, if 2% of the population carry glass with matching refractive index from the same type of source, and the match probability from the scene glass to the suspect is calculated at 2%, the LR is 1/0.02 = 50. For a fibre comparison where 5% of garments in the relevant population share the same colour and polymer type, and the match probability is 5%, the LR is 20. In a case with independent glass and fibre evidence, the combined LR is the product of the individual LRs, giving 50 x 20 = 1000, provided the two evidence types are conditionally independent given the hypotheses.

The independence assumption deserves scrutiny. Glass and fibre from the same incident may not be independent if both are explained by a single activity. The analyst must consider whether conditioning on the hypotheses makes the two observations statistically independent. When dependence is suspected, combining LRs multiplicatively overstates the total evidential value, and a joint LR must be calculated instead.

An LR is only as good as the data used to compute it. In DNA, the population frequencies are derived from large, well-characterised databases spanning thousands of profiles per locus per ethnic group. For many other forensic disciplines, the databases are smaller, less representative, or less rigorously validated. An LR derived from a database of 50 glass measurements carries far more uncertainty than one derived from 50,000.

The empirical approach to LR estimation uses a set of known same-source and known different-source comparisons to calibrate a score-based LR model. The analyst generates a comparison score from the evidence, then maps that score to an LR using the calibrated model. This approach has been applied to fingerprints, speaker comparison, handwriting, and shoemarks. The validity of the resulting LR depends on how well the calibration data represent the actual case conditions, including the population of potential alternative contributors.

Validation requires showing that the LR system is well-calibrated: when the system reports LR = 100, cases with that score should, across a large test set, show the prosecution hypothesis true approximately 100 times more often than the defence hypothesis. Poorly calibrated systems can systematically overstate or understate the LR, introducing systematic error into every case that uses them. The ISO/IEC 17025:2017 standard for testing and calibration laboratories, applied by accreditation bodies including UKAS in the United Kingdom and NABL in India, requires forensic laboratories to validate their methods, including LR calculation systems.

See also Role of Statistics in Evidence Evaluation for a broader treatment of how statistical tools fit into forensic reporting practice.