How Detective Intuition Becomes Mathematics: Modeling Forensic Trace Evidence

Prototype: One Fingernail Fragment, Ten Fingernails

In 1981, a murder in Mudanjiang, Heilongjiang left only a torn fingernail free edge inside the victim. DNA technology did not yet exist, and fingernail identification was a blank in China. Cui Daozhi initiated a study: investigators collected all ten fingernails from each suspect and compared them one by one with the crime-scene fragment. The fragment matched the right little finger of one suspect with high correspondence. The film places this experience in a 1983 case; a 1971 bite-mark comparison using a magnifying glass, which pioneered bite-mark same-source identification, also comes from Cui's real work.

Measurement: Turning "Looks Like" into Numbers

Intuition says "these two bite marks look alike." The model asks: where exactly? A bite mark can be decomposed into dental arch curvature, tooth gaps, occlusal relationship, defect positions and angles. A fingernail free edge can be decomposed into ridge count, ridge spacing, and fracture shape. Each item is quantified and recorded as a numerical feature.

Why One Fingernail Suffices: Multiplication of Features

A single matching feature proves little; many features matching simultaneously is unlikely to be coincidence. Assume each feature randomly matches a given person with probability 1/10, and features are independent. Then the probability of n features matching simultaneously is (1/10)^n. With 4 matching features, about 1,000 people in a population of 10 million would "coincidentally" match; with 8 features, only about 0.1 person. In 2020, 86-year-old Cui Daozhi spent nine continuous days comparing a half blood fingerprint that had baffled police for 35 years, finding 8 decisive feature points. Real features are not independent and have different population frequencies; each feature's prevalence must be established from large sample statistics, not guessed.

"Similar" Does Not Equal "Same Source": Two Types of Error

Two impressions from the same source are never identical: bite marks vary with biting force, fingerprints with pressure distortion. Therefore same-source similarity follows a distribution, not a perfect score; different-source similarity also follows a distribution, and the two overlap. Drawing a threshold in the overlap zone defines the decision rule. A lenient threshold increases false positives (wrongful identification); a strict threshold increases false negatives (missed identification). Using illustrative numbers: raising the threshold from 65 to 75 reduces the false positive rate from 0.6% to 0.02%, but raises the false negative rate from 3% to 27%. Forensic practice chooses to tolerate false negatives rather than false positives. When evidence is insufficient, the conclusion is "conditions for identification not met." The "zero error" record rests on this restraint in threshold setting.

Evidence Strength Depends on Search Population: Bayesian Reasoning

The model must answer an often-overlooked question: in how large a pool is the comparison made? This requires Bayes' theorem: Posterior Odds = Prior Odds × Likelihood Ratio The likelihood ratio measures the evidence's intrinsic weight: how many times more likely the true perpetrator is to leave this trace than a random person. The prior odds depend on the size of the suspect pool. The same evidence — "only 1 in 100,000 people would match this closely" — yields a posterior probability of ~99.99% when the pool is 10 suspects, but only ~1% when the pool is 10 million. Equating the random-match probability (1/100,000) directly with the posterior probability (99.999%) is the well-known "prosecutor's fallacy." The 1981 fingernail case followed this logic: investigation first narrowed the pool to a few suspects, then the physical evidence clinched the identification in that small pool. Physical evidence and investigation multiply; they are not alternatives.

No Precedent? Create One: Experimental Validation Cycle

Model parameters come from experiments. The film's line "No precedent? Create one" reflects Cui's actual approach. To understand bullet-mark regularity, he fired 3,000 rounds from the same gun, comparing minute changes on bullet heads — measuring within-source variation. From 1975, he and colleagues from four provinces spent four years collecting 125,000 fingerprint cards from 12,500 people, using mathematical statistics to establish normal values for Chinese palm regions and their relationships with height, age, and build — building the population background of "what ordinary people look like." Experiment and modeling form a loop: the model specifies what to measure; experiments supply data; data set features, thresholds, and error rates; blind testing on samples not used in modeling validates the model before casework use. Discriminability can be roughly understood as between-source difference divided by within-source variation; the higher the ratio, the more reliable the identification. Fields lacking precedents often lack precisely such data.

The Three Questions of Intuition

Returning to the three questions. How to measure intuition? Decompose into features and quantify each. How to qualify? Speak with thresholds and probabilities, and state how often you might be wrong. How to record? Write down methods and data so others can replicate and reach the same conclusion. Intuition is not superfluous — an expert's glance that "something is off here" is often the starting point for modeling. But intuition must pass through measurement, qualification, and recording to become evidence. "Passage leaves traces; but traces become evidence only after measurement, comparison, judgment, and recording. Same-source identification is essentially a mathematical model: more features → fewer coincidences; threshold governs two errors; suspect pool determines evidence weight; every model parameter comes from repeated experiments."