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The Bayesian Detective Problem – in a nutshell.

March 21, 2019

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. We know that one of these suspects is the guilty party, and we know that whoever it was acted alone.

So 20 per cent is the prior probability of guilt for each of the five possible killers, 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 four suspects is guilty rises to 25 per cent (one chance in four).

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 very reliable witnesses at the Communion service in the chapel along with Reverend Green.

So the big question is how we should 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 best estimate of the revised probability that each of Mustard, Scarlett and Plum committed the murder?

Solution

One possibility would be to 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 updating 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 estimate of 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 filter this evidence through the baseline of the probability of the hypothesis being true before we became aware of the new evidence (Mrs. Peacock’s elimination from the enquiry). This prior probability was twice as big for Colonel Mustard as for either of the other remaining suspects.

Epilogue

The estimated 50 per cent probability of guilt was more than sufficient to persuade the Crown Prosecution Service to haul the Colonel before a jury of his peers. In the event the jury convicted him, falling victim to the classic Prosecutor’s Fallacy. Like so many juries before them, they confused the probability that someone is guilty in light of the evidence with the probability of the evidence arising if they were guilty. The likelihood of Sir Caliban being shot in the library if the Colonel was guilty of murder was quite high, and this led to his conviction. Unfortunately for the Colonel, the relevant probability (that he was guilty of murder given that Sir Caliban was shot 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. He is serving a life sentence. Miss Scarlett is living as a tax exile in Belize.

 

Exercise

A murder has been committed and there are only five people who could have done it. There are no clues, np prior history that we know of. So we consider each suspect equally likely at the start of the investigation. The names of the suspects are: Reverend Green, Colonel Mustard, Miss Scarlett, Professor Plum, Mrs. Peacock.

  1. What is the prior probability of guilt for each individual suspect?
  2. Now the first clue is found, which eliminate Revd. Green. What is the new probability that each of the remaining individual suspects is guilty?

A new clue now arises which casts doubt upon the alibi of Colonel Mustard, whose probability of guilt we now judge to rise to 40 per cent.

3. What is the new probability that each of the other suspects is guilty?

The third clue now eliminates Mrs. Peacock.

4. What are the new probabilities of guilt that you, as a Bayesian, will attribute to Colonel Mustard, Miss Scarlett and Professor Plum?

 

References and Links

Books to teach yourself probability and Bayesian statistics. http://ucanalytics.com/blogs/probability-bayesian-statistics-books-self-taught/

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