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The Most Important Idea in Probability. Truth and Justice Depend on Us Getting it Right.

December 13, 2014

People have been wrongly hanged because of it. People have been wrongly punished because of it. And people have suffered unnecessarily because of it. It’s the mistaken belief that the probability of something being true based on seeing the evidence is the same thing as seeing the evidence if something is true. This can have, has had and continues to have, devastating implications. In fact, the probabilities of these two things will often diverge enormously.

To dramatize the problem I will introduce you to the Shakespearean tragedy, Othello. In the play, Othello’s wife Desdemona is set up by the evil Iago, who plants a treasured keepsake that Othello had given her in the home of young Cassio. When Othello comes upon the keepsake, he soon leaps to the mistaken conclusion that Desdemona has been unfaithful to him, with tragic consequences.

Othello made the mistake of believing that the probability that Desdemona was unfaithful given the evidence of the treasured keepsake being found in Cassio’s home was the same probability that the keepsake would have been found in Cassio’s home if Desdemona had been unfaithful to him. Easy mistake. We do it all the time in everyday life, usually with less dramatic implications. More importantly, juries do it all the time, as do practitioners in others fields, like medicine.

Let’s put it another way. What is the chance that someone who has been repeatedly shot in a flat that you rent out would die? Very high. The evidence here is the dead person, the gunshot wounds and the fact that you have access to the flat. The hypothesis is that you are the murderer. Now, the probability we would see that evidence if you ARE the murderer is 100%. But the probability that you are the murderer given that we see that evidence is much lower. There are perhaps many different people who could have committed the murder, even if you are one of them. This seems obvious, and when stated this way it IS obvious, but in real life the problem is usually not stated or understood so clearly, and is often disguised.

This is sometimes referred to as the ‘Prosecutor’s Fallacy.’ It is the fallacy of making out that someone is guilty because the evidence is consistent with their guilt. This is often enough to convict, because this  measure is often confused with the probability that the accused is guilty given that the evidence exists. They are totally different things. But even when they are clearly distinguished, the probability we assign to guilt can be seriously over-estimated because of a common cognitive failing known as the prior indifference fallacy. This is the fallacy of believing that the likelihood that something is true rather than false, when we have little prior idea, starts out as 50-50. This is just not so without proper justification but the implications of this belief, which may be implicit, are potentially huge. The prior probability, in the absence of any evidence, is simply not 50-50 unless there is a very good reason to believe that to be so before we see any evidence. Unless we can anchor this properly, all successive evidence-based reasoning will be flawed.

Fortunately for us, there is a rule used by those conversant with the laws of probability which can in fact help determine the actual relation between the truth of a hypothesis and the evidence relating to that hypothesis. The solution it arrives at is very rarely the same as would be arrived at without it. It is called Bayes’ Rule, but not many people know it, or how to apply it. Until more people do, the relationship between truth and justice is likely to remain severely strained.

Further Reading and Links

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