Bayes’ Rule and Court Ruling
We have seen in class that the use of Bayes’ Rule to determine the likelihood of a particular scenario can be very powerful and even counterintuitive in the outcome it predicts. The work on Bayes’ Rule for legal decision-making by Eiki Satake and Amy Vashlishan Murray in the Journal of Statistics Education discusses an interesting application of Bayes’ Rule in determining a verdict in court based on a sequence of presented evidence.
Bayesian methods have been developed in an effort to quantify the likelihood that a particular ruling will be made in light of presented evidence, and have seen success in real-world application; according to the paper, a model by Marshall and Wise has correctly predicted the pronouncement of 80.5% of jurors in deciding the guilt of defendants. Bayes’ Rule has also played an active role in testimonies of court cases, reaffirming its capacity to illuminate significant and perhaps unexpected realities regarding the likelihood of events. The approach to analyzing guilt through Bayesian methods, as defined by Finkelstein and Fairley in “A Bayesian Approach to Identification Evidence”, is to perceive guilt as the outcome of “an accumulation of probabilities” comprised of the evidence presented in court and its validity. In an illustration of this method, the paper presents an example in which the court is provided with 3 pieces of evidence – matches in blood type, fingerprint analysis, and DNA sample –linking a murder suspect to the crime.
Let G represent the probability that the suspect is guilty; then Pr[G|E1,E2,E3] defines the probability of guilt given the sequence of presented evidence. First, we can determine Pr[G|E1] as
When considering the first piece of evidence, Pr[G|E1] captures the probability that the suspect is guilty given that his/her blood type matches that of the sample obtained from the crime scene. Pr[E1] represents the probability that an individual possesses the blood type of the victim. The paper assumes that the individual found guilty must have a matching blood type, and thus Pr[E1│G]=1. Pr[G], the probability that the suspect is guilty prior to revealing any evidence, is a value whose overall influence in the final probability is explored at the end. Similar calculations for the following pieces of evidence result in the following relations:
The probability that the second piece of evidence is incriminating now depends on conditional probabilities derived in the previous step, which become integrated as the prior knowledge of the current “state”. This highlights the cumulative character of the calculation. Finally, we obtain that
The likelihood that a match between evidence and the suspect’s own traits is computed using statistical data such as the population carrying a particular blood type and the fidelity of fingerprint tests. However, the effect of the prior probability that the suspect is guilty, Pr[G], is interesting to observe. The paper performs calculations for several discrete values of Pr[G]; as an extension of this in a continuous plot, we can observe how Pr[G│E1,E2,E3] escalates rapidly then levels off in response to fluctuations in Pr[G]. Plotted below are these results as well as the probabilities following each additional piece of evidence presented:
Graphically it is evident that a probability of finding the suspect guilty given all pieces of evidence reaches 50% very quickly, and is as high as 97% when Pr[G]=0.05, holding all other parameters constant. The import of this variable is especially interesting to note in the progression of introducing new evidence, as it is embedded within Bayes’ Rule and subject to the remaining variables as it propagates through the calculation. We see that its relative impact is not as great when only the first piece of evidence is presented, which is reminiscent of the class example regarding a medical diagnosis of a rare condition; if the suspect is one of a large enough population, the evidence may not significantly incriminate him/her, regardless of other influential factors at play.
Source: http://www.amstat.org/publications/jse/v22n1/satake.pdf