How Bayes’ theorem was used in law to prove innocence

A very interesting case happened in 1995, Regina v Adams, where the defense used Bayes’ theorem to prove his client’s innocence.

Thomas Bayes (1701–1761)

Brief of the case: A woman named Regina was raped in London on April 1 1991, as it was during early morning hours and his attacker grabbed her from behind she wasn’t able to see him. London police immediately took vaginal swab and try to match any possible DNA fingerprints from their database, they found match with a guy named Adam.

On one side we have the prosecutor who had a highly incriminating DNA evidence, and on the other side, the defense, who decided to counter this came with a two pronged attack to prove his client is not guilty

Two pronged attack by the defense

On two fronts the defense aimed to dismantle prosecutor’s claims

a) assessing the DNA match probability: The DNA match was translated into probabilistic language as the chances the semen from vaginal swab belonged to Adam is 1 in 200 million (highly probable!). This probability was attacked by the defense, criticizing the approach on the way it was constructed and said the probability was much closer to 1 in 2 million or even lower 1 in 200,000.

b) bringing other facts into the case which favored the accused: Regina identified the man to be around 20 yrs old, while Adams was 37 at the time. Adams had an alibi, he was with his girlfriend which was also corroborated. When Adam was placed in a police lineup, Regina couldn’t identify him and also during court proceedings, she wasn’t 100% sure if Adam was the rapist.

Focus areas for prosecution and defense

This two pronged attack brought the case to an interesting standpoint: all of the evidence except DNA supported Adam’s innocence. The defense pushed the attack further by introducing Bayes theorem here, by arguing that as the jury is confused on the verdict, they can translate the evidence into probability estimates and then use Bayes theorem to finally combine and make the decision

The Bayesian approach

Formula for Bayes Theorem

The defense had an interesting problem at hand. How do you convert evidence into a probability estimate? The defense employed Peter Donnelly (who later became head of statistical science at University of Oxford) to solve this problem.

Prof Donnelly came up with a questionnaire to solve this dilemma. He mainly concentrated on three questions to get his estimates of prior odds.

  1. What is the chance, assuming nothing else about the case, that the rapist came from the local area? Donnelly here wanted to assign a probability to, in absence of any evidence, Adam being a local man. He created a 15-km radius and based on council data there were close to 150,000 men between age of 18 to 60 at that time, then he assigned a 0.75 chance that attacker is a local man. Based on this, chances of Adam being an attacker is 1 in 150,000/0.75 = 200,000.
  2. What is the chance that the victim would fail to identify Adams? Now this is an interesting question which Donnelly posed to jurors. Donnelly wanted to get an estimate of a ratio, ratio of a) if Adam was guilty, what % chance is that he would not match victim’s description b) if Adam was innocent, what % chance is that he would not match victim’s description. This ratio was used to convert the evidence that victim wasn’t able to identify Adam into a probability estimate. Donnelly got the ratio to be around 1:9
  3. What is the chance of the alibi evidence? Donnelly estimated a ratio of 1:2 for this

Leaving out the DNA evidence, the probability Adam was guilty was 1 in 3.6 million

Weighing this against the DNA evidence (1 in 200 million), the final probability estimate comes out to be 1:200,000,000/(1:3,600,000) = 1:55. The odds in favor Adam’s innocence increased drastically. Taking all the evidence into account, Donnelly claimed that it is 55 times more likely Adam is guilty than innocent (highly in favor of Adam)

In an nutshell, non-probabilistic evidence was converted into a probability estimate and bayes theorem was used to combine all the evidence and resultant odds came in favor of defendant’s innocence. In a sense, Donnelly used a simple probabilistic model to arrive at decision rather than relying on jury’s judgment (which would mostly be against the defendant considering the DNA evidence). In the end what defense was to create a reasonable doubt in jury’s minds to have the verdict as not guilty.

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