Computing Likelihood Ratios: Worked Examples

Every LR calculation begins by fixing two propositions at the same level of the hierarchy. Source-level propositions address who or what left the trace ('the defendant is the source' versus 'an unknown person is the source'). Activity-level propositions address what happened ('the defendant handled the object' versus 'the defendant never touched it'). Mixing levels in a single LR is an error: the numerator and denominator must address the same question.

The general LR formula is: LR = P(E | Hp) / P(E | Hd), where E is the complete set of observations. In simple cases, P(E | Hp) = 1 because if the defendant truly is the source, we would certainly observe the match. The LR then reduces to 1 / P(E | Hd), so estimating the denominator is the core calculation. In more complex cases, such as mixtures or partial profiles, the numerator may itself be a probability less than 1.

The verbal scale for LR values translates numerical outputs into courtroom language. The ENFSI and the UK Forensic Science Regulator's guidance use similar bands: LR from 1 to 10 is 'weak support', 10 to 100 is 'moderate support', 100 to 1000 is 'moderately strong support', 1000 to 10,000 is 'strong support', and above 10,000 is 'very strong support' for Hp. Values below 1 invert: LR of 0.1 (i.e., 10 in favour of Hd) is 'weak support for Hd'. The verbal scale is guidance, not a legal threshold.

A standard DNA profile consists of genotype observations at multiple short tandem repeat (STR) loci. At each locus, the analyst observes two alleles (or one allele if the person is homozygous). Under Hp (defendant is the source), the probability of observing a matching profile is 1. Under Hd (an unknown, unrelated person is the source), the denominator is the probability that a randomly chosen person from the relevant population has the same genotype at every locus.

For a heterozygous genotype with alleles a and b at a single locus, the genotype frequency under the Hardy-Weinberg equilibrium assumption is 2 x p(a) x p(b), where p(a) and p(b) are the allele frequencies in the reference population database. For a homozygous genotype with allele a, the frequency is p(a)^2, sometimes corrected upward using the theta (substructure) correction: p(a)^2 + p(a)(1 - p(a)) x theta. The product rule then multiplies frequencies across loci to give the random match probability (RMP). The LR = 1 / RMP.

Using the three loci in the table as a simplified example: RMP = 0.0836 x 0.0728 x 0.0234 = approximately 1.42 x 10^-4, or about 1 in 7,000. The LR = 1 / 0.000142 = approximately 7,000. Modern 15-to-20-locus profiles produce RMPs typically in the range of 1 in 10^15 to 10^20, giving LRs that are astronomically large. The random match probability concept topic covers the interpretation and misinterpretation of these figures in detail.

Glass fragments are a common form of trace evidence: a broken window leaves shards that may transfer to a suspect's clothing. The refractive index (RI) of glass, measured by a method such as temperature-controlled immersion microscopy (GRIM3), is a physical property that varies between glass sources. The LR question is: do the crime-scene glass fragments and the control glass from the defendant share a common source?

Under Hp, the questioned fragments came from the crime-scene window, so their RI measurements should be consistent with the RI of the control glass from that window. Under Hd, the fragments came from another, unknown glass source, so their RI distribution should reflect the general population of glass sources in the environment. The LR is therefore: LR = P(RI measurements | same source) / P(RI measurements | different source).

The numerator is computed from a kernel density estimate (KDE) fitted to replicate RI measurements of the control glass. Each replicate measurement from the questioned sample is evaluated at the corresponding point on that KDE density curve, giving a probability density value. The denominator is computed from a KDE fitted to a reference database of RI measurements from many different glass sources, representing what an analyst would expect to observe if the fragments came from a random glass pane. Multiplying (or summing in log space) across the questioned fragments gives the combined LR.

The Curran, Triggs, Buckleton, and Weir approach (published in 2000 and subsequently refined) formalised this KDE method and it is now implemented in software tools including the R package 'forensic'. The UK Home Office Forensic Science Service database, later maintained by the Forensic Science Regulator, contains tens of thousands of float glass RI measurements and provides the denominator distribution for cases in England and Wales. Equivalent databases exist in the Netherlands (Netherlands Forensic Institute) and Australia (AFP). LRs for glass RI comparisons in casework typically range from low values (a few tens) when the RI of the questioned glass is common in the reference population, to very high values (millions) when it falls in a thin tail of the distribution.

Forensic speaker comparison addresses the question of whether a known speaker (the suspect, whose voice is recorded in a police interview) is the same person as an unknown speaker (captured on a crime recording, a surveillance tape, or an intercepted call). The propositions are typically: Hp = 'the suspect spoke the questioned recording'; Hd = 'a different person, drawn from the relevant population of speakers, spoke the questioned recording'.

Automatic speaker recognition (ASR) systems extract feature vectors from the recordings, typically mel-frequency cepstral coefficients (MFCCs) representing the spectral envelope of speech, and compare the suspect and questioned recordings using a statistical model. The most widely used current architecture is the probabilistic linear discriminant analysis (PLDA) model applied to speaker embeddings (i-vectors or x-vectors). The system outputs a score: a high score indicates acoustic similarity, a low score indicates dissimilarity.

That raw score is not yet an LR. It must be calibrated: the score is mapped to a log-LR by evaluating it against a background population of same-speaker and different-speaker score distributions derived from a reference corpus. If the score falls where same-speaker pairs typically cluster, the log-LR is positive and large; if it falls where different-speaker pairs cluster, the log-LR is negative. Calibration is performed using logistic regression or the Pool Adjacent Violators (PAV) algorithm. The ENFSI Speaker Identification Working Group has published guidelines specifying validation requirements: a calibrated system must demonstrate adequate performance on a representative validation set before its outputs are used in court.

An LR of exactly 1 means the evidence is equally probable under both propositions and provides no discriminating information. This is a neutral result, not an absence of evidence. An LR of 1,000 means the evidence is 1,000 times more probable under Hp than Hd. An LR of 0.001 means the evidence is 1,000 times more probable under Hd than Hp: the finding actively supports the defence.

A common error is to report only LRs above 1 and treat LRs below 1 as non-results. This is a form of selective reporting that distorts the picture for the court. ENFSI guidelines, the UK Forensic Science Regulator's Codes of Practice and Conduct, and the ILAC G19:08/2014 guidelines on forensic science laboratories all require that LR calculations be reported regardless of their direction. A scientist who discovers the evidence supports the defence has an obligation to report that finding.

The precision of an LR estimate also matters. Point estimates of LR are rarely exact: they depend on database choices, model assumptions, and the variability of the measurements. Some laboratories report confidence intervals or sensitivity analyses alongside the central LR value. The Bayesian credible interval approach and the bootstrapped confidence interval are both used in practice. A reported LR of 1,000 with a 95% confidence interval of 200 to 5,000 communicates a different level of certainty than a point estimate without bounds.

An LR model must be validated before it is used in casework. Validation tests the model on a known dataset: for DNA, this means computing LRs for profiles where ground truth is known (same-source and different-source pairs) and checking that the model assigns high LRs to same-source pairs and low LRs to different-source pairs. The Tippett plot, which graphs the cumulative distribution of LRs for same-source and different-source pairs on the same axes, is the standard validation visualisation. A well-calibrated model shows the two curves separating cleanly.

Empirical cross-entropy (ECE) is the standard scalar metric for evaluating LR system performance. It measures how much information is lost (relative to a perfect system) when the model's log-LR outputs are used as Bayesian evidence. Lower ECE is better. The Cllr (log likelihood ratio cost) is an equivalent metric used in speaker comparison literature. Both measure the combination of discriminating power and calibration: a model that discriminates well but is miscalibrated still has poor Cllr.

Courts in multiple jurisdictions have scrutinised LR models. The New Zealand Court of Appeal in R v Pengelly (1992) addressed the admissibility of statistical DNA evidence. The UK Court of Appeal in R v Deen (1994) identified the prosecutor's fallacy in a DNA case and clarified the distinction between the RMP and the probability of guilt. In India, under the Bharatiya Sakshya Adhiniyam 2023 (which replaced the Indian Evidence Act 1872), expert opinion is admissible under Section 39 when the subject requires specialised knowledge; the LR model and its validation evidence would typically be disclosed to the defence as part of the expert's report. Equivalent disclosure obligations exist under the US Federal Rules of Evidence Rule 702, and under the EU directive on minimum standards for criminal proceedings.