The second in the list of ten of David Sumpter’s The Ten Equations that Rule the World is called the judgement equation. It concerns the need to consider models, including the perceptions inside our heads, and alternatives, before making judgment calls. If M represents the model and D represents the data, his equation is of the following form.
Ladies and gentlemen, Sumpter’s second equation is nothing other than the very foundation of this blog site—the Bayes’ Equation. He uses examples involving objective measures and subjective experiences in life to emphasise an essential requirement for quality judgements—updating the thought process with supporting and opposing information.
His first example discusses the anxieties of an air traveller about a potential crash while experiencing an unexpected shaking of the plane. He imagines that since all crashes are associated with massive shakes, this instance will also lead to a calamity. Instead, the mathematician in him must evaluate the situation carefully.
This is a case where all the required information is well documented. To calculate the probability of the model (the crash) given the data (the shake), he can use the judgement equation using well-documented data from the airline industry.
The probability of a crash is about 1 in 10 million. The probability of shakes given a crash is 1 (the flight almost always shakes before a crash). Then comes the most crucial point. Many non-crash events also lead to shakes. These are your countermodels. He estimates the probability of shaking, given that there is no crash, is 1 in 100.
The second case examines whether Amy should abandon her friend (Rachel) when she overhears Rachel talking nasty things about Amy with another friend. Amy has no quantitative information on friendships and nastiness, unlike the airline database. She estimates:
The prior probability of a person being nasty is 1 in 20; P(M) = 1/20
50% of the time, a nasty person talks nonsense about friends; P(D|M) = 1/2
Rachel just had a bad day is 10%; P(D|Mc) = 1/10
There is a 4 in 5 chance that Amy is a decent girl!
The judgment equation tells us to be slow and structured before reaching a verdict. Each piece of information or advice allows one to build models and alternatives.
