Some Bayesian thoughts on the classic mystery genre, prompted by watching on Netflix episodes of the Poirot series with David Suchet (which is really excellent by the way).

A common pattern in classic mystery stories is that there is a an obvious suspect, who had clear motive, means and opportunity for the crime (perhaps there is also some physical evidence against him/her). However, there is one piece of evidence that is unexplainable if the obvious person did it: a little clue unaccounted for, or perhaps a seemingly inconsequential lie or inconsistency in a witness' testimony. The Great Detective insists that no detail should be ignored, that the true explanation should account for all the clues. He eventually finds the true solution, which perfectly explains all the evidence, and usually involves a complicated plot by someone else committing the crime in such a way to get an airtight alibi, or to frame the first suspect, or both.

In Bayesian terms, the obvious solution has high prior probability P(H), and high P(E|H) for all components of E except for one or two apparently minor ones. The true solution, by contrast, has very high probability P(E|H) for all components of E. It is also claimed by the detective to have high prior P(H) (the guilty party tends to be someone with an excellent motive, they just had been dismissed as a suspect because of a seemingly perfect alibi). However, there is here a required suspension of disbelief, in that in real life there is a very low prior probability of someone plotting a crime (and successfully carrying it out) with a convoluted, complicated plot in order to get an alibi. In real life, the detective's solution would be dismissed because of a low P(H), and the detective's insistence on finding a solution that maximizes P(E|H) at the cost of P(H) would be flawed from the point of view of Bayesian rationality.