Extreme Opinions and Bayes’ Theorem
As a society, we are taught to be open-minded so that we may better understand others’ ideas and points of view. This way, we may be more tolerant of each other’s beliefs and act more empathetically toward each other. However, extreme opinions abound, especially in politics. If you look at a presidential candidate’s Facebook page and search through the comments on the posts, you will see enthusiastic supporters and scathing critics. The same is true for YouTube comments under a video that expresses a controversial opinion. People with extreme opinions often do not change their minds, even if they are presented with strong refuting evidence. Part of the reason for stubbornness is psychological: people do not like to admit that they are wrong. However, Bayes’ theorem suggests that there may be some rational basis to holding a belief that contradicts the evidence.
Bayes’ theorem is given by the equation Pr[A | B] = Pr[B | A] Pr[A] / Pr[B]. This equation encodes the idea that given the prior probability of an event, one can update the probability as more evidence is collected. The prior probability is Pr[A], the updated probability is Pr[A | B], and the evidence is represented as Pr[B | A] / Pr[B]. A Live Mint article describes the application of Bayes’ rule to a game of cricket. In cricket, bowling pitches and batting pitches have different probabilities of producing a given sequence of scores. Given a sequence of observed scores in a game, what is the probability that a bowling pitch or a batting pitch was conducted? Suppose that person 1 has a prior belief that there is a 50% chance that a batting pitch occurred. After observing a sequence of scores, this probability is updated to 85%. Suppose that person 2 has a prior belief that there is a 10% chance that a batting pitch occurred. After seeing the same sequence of scores, this probability is updated to 38%. I will not show the calculations here, but these results indicate that given the same evidence, people with different opinions about the prior probability of an event may reach different conclusions. In particular, person 1 is reasonably convinced that a batting pitch occurred, but person 2 is not convinced.
Bayes’ theorem rationalizes the notion that an extreme opinion may be unswayed by evidence. Suppose that a man holds the extreme opinion that goblins live inside vending machines and are responsible for dispensing products. He is stubborn enough to believe that there is a 0% chance that his opinion is wrong. Let us analyze this situation with Bayes’ theorem. Let event A be the prior probability that his opinion is wrong, and let event B be evidence against his opinion. We know that Pr[A] = 0, so Pr[A | B] = Pr[B | A] Pr[A] / Pr[B] = Pr[B | A] * 0 / Pr[B]. As long as Pr[B] ≠ 0, this expression reduces to 0. Therefore, no matter what evidence you throw at him, he will conclude that there is still a 0% chance that his opinion is wrong. Bayes’ theorem suggests that when people are absolutely sure or almost totally sure of a belief, it will be difficult or impossible to convince them otherwise.