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Who shot Sir Caliban? A Case for the Bayesian Detective.

April 8, 2017

A murder has been committed. There are five suspects, all of whom we consider equally likely to be guilty at the start of the investigation.

So 20 per cent is the prior probability of guilt for each suspect, before any new evidence is found. The names of the suspects are: Reverend Green, Colonel Mustard, Miss Scarlett, Professor Plum and Mrs. Peacock. The codename for the murder investigation is Operation Cluedo. The victim was Sir Caliban Mackenzie, a famed anthropologist, who was shot in the library while examining a rare first edition of Newton’s Principia.

Four hours into the investigation, evidence turns up which eliminates Reverend Green. He was leading the Holy Communion Service in the chapel at the time of the murder. There are now four remaining suspects, and so the probability that each of the remaining suspects is guilty rises to 25 per cent.

Two hours later, a new clue now arises which casts some doubt on the alibi of Colonel Mustard, whose probability of guilt we now judge to rise from 25 per cent to 40 per cent.

As a result, the probability that one of the other three suspects is guilty falls by 15 per cent, down from a total of 75 per cent to 60 per cent. Since each of the three is equally likely to be guilty, we can now assign each a probability of guilt of 20 per cent, down from 25 per cent.

After a further 45 minutes, a third clue emerges, which eliminates Mrs. Peacock. She had been spotted by a number of reliable witnesses at the Communion service in the chapel along with Reverend Green.

So the big question is how should we now adjust the probabilities that Colonel Mustard, Miss Scarlett and Professor Plum pulled the trigger?

In other words, now that Mrs. Peacock has been eliminated, and taking account of the evidence which doubled the original likelihood that Colonel Mustard wielded the murder weapon (to 40 per cent), what is the revised probability that each of Mustard, Scarlett and Plum committed the murder?

 

Spoiler Alert: The Solution

 

One possibility would be take the 20 per cent probability of guilt we had previously attached to Mrs. Peacock, and divide this equally between the three remaining suspects.

But to do so would be wrong, and notably at variance with the toolkit of a Bayesian detective, i.e. a detective who conducts investigations using the Bayesian approach to evidence and probability.

The Bayesian approach to detective work tells us always to consider the prior probability that each suspect is guilty before deducing the probability after some new evidence is brought to bear on it. Applying this method, the correct way to adjust the probabilities attached to the remaining suspects is to do so in a way that is proportional to their prior probability of guilt before Mrs. Peacock was eliminated from the enquiry.

Since Colonel Mustard was the prime suspect, with a probability of guilt of 40 per cent before Peacock’s elimination (compared to 20 per cent for Miss Scarlett and Professor Plum), a good Bayesian needs to increase the probability we assign to his guilt by twice as much as we increase theirs. So we should now raise the probability that Colonel Mustard shot Sir Caliban from 40 per cent to 50 per cent, while we should increase the probability we assign to Miss Scarlett and Professor Plum from 20 per cent to 25 per cent.

This is all derived from Bayes’ Theorem, which tells us that in order to calculate the probability of a hypothesis being true given new evidence, we must multiply by the prior probability of the hypothesis being true before we are aware of the new evidence (Mrs. Peacock’s elimination from the enquiry). This prior probability is twice as big for Colonel Mustard as for either of the other remaining suspects.

Epilogue:

The 50 per cent probability of guilt was more than sufficient to persuade the Crown Prosecution Service to haul the hapless Colonel before a jury of his peers. In the event they convicted him, falling victim to the classic Prosecutor’s Fallacy, which is to confuse the probability that someone is guilty given the evidence with the probability of the evidence arising if someone is guilty. The likelihood of the gunshot in the library if the Colonel was guilty was quite high, and this led to his conviction. Unfortunately for him, the relevant probability (that he was guilty given the gunshot in the library) was rather smaller but bypassed in the jury’s deliberations.

Meanwhile, the actual killer, Miss Scarlett, got away scot-free. She had concealed an incriminating letter in the Principia, thinking it would be safe there, until Sir Caliban unhappily chanced upon it. This left her no option, in her mind, but to use the pistol hidden in the Georgian chest of drawers gracing the back wall of the library.

The Colonel’s appeal was unanimously rejected.

 

Further Reading and Links

https://selectnetworks.net/

 

 

 

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