Last week I had the pleasure to attend a talk by Leila Schneps on the mathematics of crime. A self-declared pure mathematician, Schneps recently became involved in studying the mathematics in criminal trials. Rather than focusing on the mathematics of forensic data, such as DNA, the talk was on the use of Bayesian networks in crime evidence analysis.
One of the most challenging tasks for juries in criminal trials is to weigh the impact of different pieces of evidence which may be dependent on each other. In fact, studies have shown that there are several fallacies that jury members fall into on a regular basis, such as overestimating the weight of a single piece of DNA evidence. This is because most people assume that the probability of an event A happening given that B happens [P(A|B)] is equal to an event B happening, given that A happens [P(B|A)]. However, this is NOT true: the two are connected by Bayes’ Rule.
For example, a forensic specialist may say that a piece of DNA is found in 1 in every 1000 people. The jury will take this to mean the the suspect must be guilty as he is that person out of 1000 (the chances of it being anyone else is so low). However, this is critically not true, as explained above.
Thus, Bayesian networks are a powerful tool to assess the weight of different kinds of evidence, taking into account their dependencies on one another, and what effect they have on the guilt of the suspect.
What are Bayesian Networks?
Bayesian Networks are a type of statistical model, which organise a body of knowledge in any given area, in this case evidence of a criminal trial, by mapping out cause-and-effect relationships and encoding them with numbers that represent the extent to which one variable is likely to affect the other. These networks are named after British mathematician Reverend Thomas Bayes, due to their reliance on Bayes’ formula. This rule can be extended to multiple variables and multiple states, allowing complicated probabilities to be calculated.
To illustrate an easy example, consider this Bayesian Network:
Let us say that the weather can only have three states: sunny, cloudy or rainy, that the grass can only be wet or dry, and that the sprinkler can be on or off. The arrows, which represent dependence, have been drawn in that way because if it rainy, then the lawn will be wet, but if it is sunny for a long time then this will cause us to turn on the sprinklers, and hence the lawn will also be wet.
By imputing probabilities into this network that reflect the reality of real weather, lawn and sprinkler-use behaviour, it can be made to answer questions like:
“If the lawn is wet, what are the chances it was caused by rain or by the sprinkler?”
“If the chance of rain increases, how does that affect my having to budget time for watering the lawn?”
In her presentation, Leila Schneps talked briefly about a book she had released entitled ‘Math on Trial’, which describes 10 trials spanning the 19th century to today, in which mathematical arguments were used (and greatly misused) as evidence. The cases discussed include “Sally Clark, who was accused of murdering her children by a doctor with a faulty sense of calculation; of nineteenth-century tycoon Hetty Green, whose dispute over her aunt’s will became a signal case in the forensic use of mathematics; and of the case of Amanda Knox, in which a judge’s misunderstanding of probability led him to discount critical evidence – which might have kept her in jail.”
After hearing Schneps transmit her passion and excitement, I am fascinated with this subject and can’t wait to get my hands on this book to learn more! M x