Bayes's Theorem
Bayes's Theorem provides a systematic method for assessing different types of information to make a decision. Robertson and Vignaux (1995) provide a hypothetical example of a jury considering a man charged with drunk driving.
The prosecution argues that 100 of every 10000 drivers stopped for traffic offenses are drunk. Ninety of them including the accused fail the coordination tests used in the field by the police and have drunk driving added to the original charge (10 drunks manage to pass the test and are not charged). The prosecutor asserts that the person on trial is part of the group of 90 who failed the test and was driving drunk. The defense responds that all this may be true, but that his client is a member of the larger, innocent, group of 180 of the 9900 sober drivers, who because they are elderly, nervous, or poorly coordinated, fail the test and are mistakenly charged. During deliberation, a mathematically astute juror points out that 10,000 stops break down as follows:10 drunks passed coordination the test and were released 90 drunks failed the coordination test and were charged 180 sober drivers failed the coordination test and were charged 9700 sober drivers passed the test and were released
That is, 66.67% of those charged are innocent—high enough to constitute "reasonable doubt.” The embarrassed prosecutor insists that in future the police also use the newest breathalyzers, which leads to the following results:
10 drunks passed the coordination test and were released 90 drunks failed the coordination test, of whom 85 failed the breathalyzer test and were charged 180 sober drivers failed the coordination test, of whom 179 passed the breathalyzer test and were released 9700 sober drivers passed the test
Now, 85 out of 86 of those charged actually were drunk, although one unfortunate is not. Each juror still must decide for himself whether a 1.17% chance of error constitutes “reasonable doubt” and vote guilty or not guilty accordingly.
The example illustrates the logic behind Bayes’s Theory, which provides an objective mechanism for weighing and aggregating different kinds of information, in this case providing jurors with a way of fulfilling their responsibility as finders of fact. In a Bayesian approach, each new experiment either increases or decreases the probability that a theory is true. This seems an improvement on the traditional dichotomy, in which each experiment either refutes a theory or leaves it standing (Chapter 1). (Bolstad 2004, Churchman & Book 1980, Gelman et. al. 2003, Winkler 2003, Yudowsky 2003).