Miscellaneous Monday—Number Twenty-three

JOHN DICKSON CARR, the grand master of the locked room mystery, wrote that in the classical detective story the author and the reader engage in what he termed "The Grandest Game in the World," as the writer attempts to outwit the reader at every turn and the reader tries to do the same to the author and reach the correct solution before the story ends. Today's academician, however, takes the Grandest Game to a whole other level . . .

"Bayesian Thought in Early Modern Detective Stories: Monsieur Lecoq, C. Auguste Dupin and Sherlock Holmes."
By Joseph B. Kadane (born 1941).
First appearance: Statistical Science, May 2009.
Article (7 pages).
Online at arXiv (HERE) (PDF).
(Note: SPOILERS for "The Murders in the Rue Morgue" and "The Purloined Letter").
(Further note to mathophobes: There is some mathematics in this paper, but don't panic! There's not enough to obscure the author's meaning.)

"Both detective stories and Bayesian analysis have flourished in the intervening century. They share some common roots."

A professional statistician shows how the forerunners of modern detective fiction made use of Bayesian theory to activate their plots—without, of course, knowing that's what they were doing.

Abstract:
"This paper reviews the maxims used by three early modern fictional detectives: Monsieur Lecoq, C. Auguste Dupin and Sherlock Holmes.
It find [sic] similarities between these maxims and Bayesian thought.
Poe’s Dupin uses ideas very similar to Bayesian game theory. Sherlock
Holmes’ statements also show thought patterns justifiable in Bayesian
terms."

1. Introduction:
   "This essay aims to examine the pattern of thought used by their respective detectives: Monsieur Lecoq, C. Auguste Dupin and Sherlock Holmes.  . . . 
[T]here is a sense in which understanding fictional characters is easier than understanding real ones. There is a fixed body of written work, and this is all the evidence there will ever be. Those words tell what characteristics of the detectives the author considers most important. When the author writes
about the way such characters go about their work, this can be taken to be authoritative."

2. Emile Gaboriau's Monsieur Lecoq:
   "Little is written about Lecoq’s methods, except for one refrain that
occurs three times: 'Always suspect that which seems probable; and
begin by believing that which appears incredible,' 'Distrust all circum-
stances that seem to favor your secret wishes,' and 'Always distrust
what seems probable!' Taken together they suggest a tinge of paranoia, perhaps  . . ."

3. Edgar Allan Poe's C. Auguste Dupin:
   "What is important to us in these three stories is the theory Poe promulgates as to how Dupin is thinking about the puzzles he sets himself to solve."  . . .  [Speaking as a statistician, our author congratulates Poe on making] "an  important and subtle point, one that it took the medical profession another century to incorporate, via the use of controlled clinical trials."

4. Sir Arthur Conan Doyle's Sherlock Holmes:
   "In contrast to Gaboriau’s single (or perhaps double) book and Poe’s three short stories, Doyle gives us four Sherlock Holmes novels and 56 short stories. So we have in one sense a great deal of information. However, Doyle seems less anxious than Poe to show us how Holmes is thinking about his tasks. When he does so, on occasion those thoughts are often reminiscent of ideas already in Poe’s stories."  . . . [Through his character] "Doyle (Holmes) is saying that predicting subsequent from preceding events is relatively straight-forward, but the reverse is hard. And this is exactly what Bayes’ Theorem does. However, that theorem is even more evident in what we must take as Holmes’ slogan, as it is repeated four times in the work. 'When you have eliminated the impossible, whatever remains, however unlikely, must be the truth.' Thus Sherlock Holmes is using, and insisting upon, Bayesian results to explain his actions."

Artwork by Ronald Searle

5. Conclusion (in which the author cross-examines himself):
  ~ "How would you describe Sherlock Holmes’ use of Bayesian ideas?"
    "Holmes certainly seems to understand the ideas, and how to use them."
  ~ "The idea is that if I am playing a game against you, my main source of uncertainty is what you will do. As a Bayesian I have probabilities on what
you will do, and can use them to calculate my maximum expected utility choice, which is what I should choose."
    "Is this consistent with what Poe writes about games?"
    "Very much so."
  ~ "Is there anything that Poe writes about games that is inconsistent with your theory?"
    "No. I think Poe understood skill in games very well, both in how Dupin outwits Minister D., and in his general introduction. As I explained earlier,
I disagree with him about chess, but as a general matter, his view of skill
in games is very similar to the one in our papers."

6. References

Typos: 'The Murders in the Rue Morgue' (1944); "as the lay progresses."

Resources:
- Wikipedia has a short article about Joseph B. Kadane (HERE).

- Wikipedia also has quite a few related articles, four of which are "Bayesian probability" (HERE), "Bayes' theorem" (HERE), "Bayesian inference" (HERE), and "Cromwell's rule" (HERE); another one that relates more closely to our article is "Bayesian game" (HERE):

  "In game theory, a Bayesian game is a game in which the players have incomplete information on the other players (e.g., on their available
strategies or payoffs), but they have beliefs with known probability distribution."

If you think of "the players" as Holmes and Moriarty (or any detective and suspect for
that matter), then you have an inkling of how Bayesian game theory could be applied
to mysteries.
- There's no escaping Poe's influence on detective fiction (HERE).