The Devotion of Suspect X | How Bayes’ Rule and Decision-Making involving our life
As speculative fiction, The Devotion of Suspect X incorporates many knowledge about networks, mathematics, and probability theory into the plot design. As mentioned in the article, a discussion about the problem of P vs. NP comes up in Ishigami and Yukawa’s communication. It seems theoretically possible for a problem to be in P but not in NP since solving it is more complicated than checking the answer. However, mathematics and inference are not only found between mathematicians and physicists, but when we discuss how people perceive and decide a problem, we can also apply Bayes everywhere.
In the fiction, the police ask the mother and daughter where they were on the night of the murder, and the daughter answers she went to a movie. There are three actions that Ishigami asked the mother and daughter to do:
1. Don’t take out the movie ticket.
2. When the police asked for it, pretending not to find those.
3. Put the movie tickets in the movie playbill and show at the end.
How could we use Bayes’ rule to determine whether the mother and daughter kill someone on the night of the murder? And how could those actions deceive the police by probability? As a mathematician, Ishigami is a master at lying, but how probability theory works here? We can use Bayes’ rule to tell the truth.
If we make event A equal to “mother and daughter killed someone,” and event B equal to “mother and daughter have movie tickets,” we could set up the probability by:
P(A): the probability that the mother and daughter went to kill someone
P(B): the probability that the mother and daughter have movie tickets in their hands
P(A|B): the probability that the mother and daughter went to kill someone, given the movie tickets were found in their hands
P(B|A): the probability that the movie tickets were found in their hands, given the mother and daughter went to kill someone
The mother and daughter try to perjure themselves with the movie tickets, which means that they want to make the value of P(A|B) smallest in the mind of the police. At this point, we have the Bayes’ formula obtained in class: P(A|B) = P(B|A) * P(A)/P(B)
Since the purpose of the mother and daughter is to make P(A|B) smaller, we can see from the above formula that this can be done by reducing P(B|A) on the right side of the equation, which means the probability that they would have movie tickets in their hands (event B), given if the mother and daughter really went to kill someone last night (event A).
What Ishigami did was to let the mother and daughter not produce evidence easily. Deliberately, Ishigami made them look like they can’t give evidence that makes people more and more convinced of how small the probability is that they can still provide evidence B after event A has happened – which is P(B|A). Most people are initially confronted with that it is not difficult to get a fake ticket even if you went to kill someone. This assumption implies an expectation of the probability of “killing someone and getting a fake ticket.”
So what is the benefit of pretending not to have a ticket and then subsequently taking them out?
In terms of probability, the mother and daughter redirect the police’s consideration, and make them think — it is not easy to get a fake ticket — by not showing the tickets at the beginning. This action lowers P(B|A), which reduces the police’s expectation of the probability of “killing someone and getting a fake ticket.” At the same time, when the mother and daughter can not find the ticket at first, the police will subconsciously raise P(A), that is, the probability of the mother and daughter going to kill someone. And next, having the tickets will make the police feel that their guess was wrong. Look back to the Bayesian formula P(A|B) = P(B|A) * P(A)/P(B), everything they did in reducing the value of P(B|A) and P(A), also let the left side – P(A|B) – is also significantly reduced.
The example in fiction is one of the ubiquitous use of Bayes’ Rule, a decision-making model under uncertainty involving life and art, where just a few small tricks can change the base and conditional probabilities in people’s minds.
Source:
https://www.ft.com/content/22c6ab18-a972-11e0-bcc2-00144feabdc0
https://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf974