Bayes’ Theorem was named after Reverend Thomas Bayes who first provided the equation that would later be developed by Pierre-Simon Laplace to give the conditional probability equation. It describes the probability of an event occurring, taking into consideration prior knowledge of the event and how well the event fits in with the new evidence. This reflects the theorem at its very core: initial belief + new data ? improved belief.1 One of the many applications of Bayes’ theorem is Bayesian inference, which allows one to better form an interpretation of a situation as any subjective degree of belief should be altered to take into account for other related evidence. Bayesian Inference is one of the elements that sets Bayes’ theorem apart from the usual conditional probability. Bayes’ theorem has had its controversies in the past and even today2, reasons ranging from it being biased to it being too uncalculated2. Some may even fall into the trap of taking the frequentist approach, which is when one only looks at the new evidence to conclude whether or not an event has taken place.
3 However, it cannot be denied that many important historical breakthroughs were made possible because of this theorem. Examples of this are Alan Turing breaking the enigma code during WWII3 and Nate Silver successfully predicting the outcome of the presidential vote in all 50 states3 during the 2012 United States elections. It has also been frequently used in the medical field to calculate probability of contracting a disease and in courtrooms. I was first intrigued to explore conditional probability in Bayes Theorem when I read about the infamous case of Sally Clark, a woman who was accused of murdering her own two children when in actual fact, they both coincidentally died of SIDS (Sudden Infant Death Syndrome).
4 This unfortunate accusation took place because prosecutors had utilized conditional probability wrongly, and like most people today, had taken the frequentist approach. The prosecution was based on the chance that 2 children from the same family dying from cot-death was 1 in 73 million thus Sally was convicted. This is known as the Prosecutor’s Fallacy5, the assumption that prior probability of random match is equal to probability that defendant is guilty.
Further research into the Prosecutor’s Fallacy led me to many other cases of false accusations just because its conditional probability had been used wrongly. For instance, a common mistake is simply forming causational links (e.g: if a man has a history of domestic abuse then he is a murderer too) while dismissing the probabilities of how rare the outcome actually is.
This fallacy will be further explored later on, and a method to help avoid this using Bayes’ Theorem will be shown. On the other hand, as Bayes’ Theorem has had many controversies, I will also be exploring how it hinders progress in courtroom and the dangers of falling into the trap of utilizing it incorrectly. That being said, an alternative to Bayes’ Theorem, specifically Hypothesis Testing will be briefly explained.