# Bayes and the Othello Problem – in a nutshell.

The majestic tragedy, Othello, was written by William Shakespeare in about 1603. The play revolves around four central characters: Othello, a Moor who is a General in the Venetian army; his beloved wife, Desdemona; his loyal lieutenant, Cassio; and his trusted ensign, Iago.

A key element of the play is Iago’s plot to convince Othello that Desdemona is conducting an affair with Cassio, by planting a treasured keepsake Othello gave to Desdemona, in Cassio’s lodgings, for Othello ‘accidentally’ to come upon.

We playgoers know she is not cheating on him, as does Iago, but Othello, while reluctant to believe it of Desdemona, is also very reluctant to believe that Iago could be making it up.

If Othello refuses to contemplate any possibility of betrayal, then we would have a play in which no amount of evidence, however overwhelming, including finding them together, could ever change his mind. We would have a farce or a comedy instead of a tragedy.

A shrewder Othello would concede that there is at least a possibility that Desdemona is betraying him, however small that chance might be. This means that there does exist some level of evidence, however great it would need to be, that would leave him no alternative. If his prior trust in Desdemona is almost, but not absolutely total, then this would permit of some level of evidence, logically incompatible with her innocence, changing his mind. This might be called ‘Smoking Gun’ evidence.

On the other hand, Othello might adopt a more balanced position, trying to assess the likelihood objectively and without emotion. But how? Should he try and find out the proportion of female Venetians who conduct extra-marital affairs? This would give him the probability for a randomly selected Venetian woman but no more than that. Hardly a convincing approach when surely Desdemona is not just an average Venetian woman. So should he limit the reference class to women who are similar to Desdemona? But what does that mean?

And this is where it is easy for Othello to come unstuck. Because it is so difficult to choose a prior probability (as Bayesians would term it), the temptation is to assume that since it might or might not be true, the likelihood is 50-50. This is known as the ‘Prior Indifference Fallacy’. Once Othello falls victim to this common fallacy, any evidence against Desdemona now becomes devastating. It is the same problem as that facing the defendant in the dock.

Extreme, though not blind, trust is one way to avoid this mistake. But an alternative would be to find evidence that is logically incompatible with Desdemona’s guilt, in effect the opposite of the ‘Smoking Gun.’ The ‘Perfect Alibi’ would fit the bill.

Perhaps Othello would love to find evidence that is logically incompatible with Desdemona conducting an affair with Cassio, but holds her guilty unless he can find it. He needs evidence that admits no True Positives.

Lacking extreme trust and a Perfect Alibi, what else could have saved Desdemona?

To find the answer, we can turn to Bayes’ Theorem.

The (posterior) probability that a hypothesis is true after obtaining new evidence, according to the a,b,c formula of Bayes’ Theorem, is equal to:

ab/[ab+c(1-a)]

a is the prior probability, i.e. the probability that a hypothesis is true before you see the new evidence. b is the probability you would see the new evidence if the hypothesis is true. c is the probability you would see the new evidence if the hypothesis is false.

In the case of the Desdemona problem, the hypothesis is that Desdemona is guilty of betraying Othello with Cassio. Before the new evidence (the finding of the keepsake), let’s say that Othello assigns a chance of 4% to Desdemona being unfaithful.

So a = 0.04

The probability we would see the new evidence (the keepsake in Cassio’s lodgings) if the hypothesis is true (Desdemona and Cassio are conducting an affair) is, say, 50%. There’s quite a good chance she would secretly hand Cassio the keepsake as proof of her love for him and not of Othello.

So b = 0.5

The probability we would see the new evidence (the keepsake in Cassio’s lodgings) if the hypothesis is false is, say, just 5%. Why would it be there if Desdemona had not been to his lodgings secretly, and why would she take the keepsake along in any case? It could have been stolen and ended up there, but how likely is that?

So c = 0.05

Substituting into Bayes’ equation gives:

Posterior probability = ab/[ab+c(1-a)] = 0.294.

So, using Bayes’ Rule, and these estimates, the chance that Desdemona is guilty of betraying Othello is 29.4%, worryingly high for the tempestuous Moor but perhaps low enough to prevent tragedy. The power of Bayes here lies in demonstrating to Othello that the finding of the keepsake in the living quarters of Cassio might only have a 1 in 20 chance of being consistent with Desdemona’s innocence, but in the bigger picture, there is a less than a 3 in 10 chance that she actually is culpable.

If this is what Othello concludes, the task of the evil Iago is to lower c in the eyes of Othello by arguing that the true chance of the keepsake ending up with Cassio without a nefarious reason is so astoundingly unlikely as to merit an innocent explanation that 1 in 100 is nearer the mark than 1 in 20. In other words, to convince Othello to lower his estimate of c from 0.05 to 0.01.

The new Bayesian probability of Desdemona’s guilt now becomes:

ab/[ab+c(1-a)]

a = 0.04 (the prior probability of Desdemona’s guilt, as before)

b = 0.5 (as before)

c = 0.01 (down from 0.05)

Substituting into Bayes’ equation gives:

New probability = 0.676 = 67.6%.

So, if Othello can be convinced that 5% is too high a probability that there is an innocent explanation for the appearance of the Cassio – let’s say he’s persuaded by Iago that the true probability is 1% – then Desdemona’s fate, as that of many a defendant whom a juror thinks has more than a 2 in 3 chance of being guilty, is all but sealed. Her best hope now is to try and convince Othello that the chance of the keepsake being found in Cassio’s place if she were guilty is much lower than 0.5. For example, she could try a common sense argument that there is no way that she would take the keepsake if she were actually having an affair with Cassio, nor be so careless as to leave it behind. In other words, she could argue that the presence of the keepsake where it was found actually provides testimony to her innocence. In Bayesian terms, she should try to reduce Othello’s estimate of b. What level of b would have prevented tragedy? That is another question.

William Shakespeare wrote Othello about a hundred years before the Reverend Thomas Bayes was born. That is true. But to my mind the Bard was always, in every inch of his being, a true Bayesian. Othello was not, and therein lies the tragedy.

**Appendix**

In the case of the Othello problem, the hypothesis is that Desdemona is guilty of betraying Othello with Cassio. Before the new evidence (the finding of the keepsake), let’s say that Othello assigns a chance of 4% to Desdemona being unfaithful.

So P (H) = 0.04

The probability we would see the new evidence (the keepsake in Cassio’s lodgings) if the hypothesis is true (Desdemona and Cassio are conducting an affair) is, say, 50%.

So P (EIH) = 0.5

The probability we would see the new evidence (the keepsake in Cassio’s lodgings) if the hypothesis is false is, say, just 5%.

So P (EIH’) = 0.05

Substituting into Bayes’ Theorem:

P (HIE) = P (EIH). P (H) / [P (EIH) . P(H) + P (EIH’) . P(H’)]

P (HIE) = 0.5 x 0.04 / [0.5 x 0.04 + 0.05 x 0.96]

P (HIE) = 0.02 / [0.02 + 0.048] = 0.294

Posterior probability = 0.294.

So, using Bayes’ Rule, and these estimates, the chance that Desdemona is guilty of betraying Othello is 29.4%.

If P (EIH’) = 0.01

The new Bayesian probability of Desdemona’s guilt now becomes:

P (HIE) = 0.5 x 0.04 / [0.5 x 0.04 + 0.01 x 0.96]

P (HIE) = 0.02 / (0.02 + 0.0096) = 0.02 / 0.0296 = 0.676

Updated probability = 0.676 = 67.6%.

**Exercise**

Othello believes that there is just a 5 per cent chance that his wife, Desdemona, would be unfaithful to him. Until he comes upon one of her treasured keepsakes in Cassio’s lodgings. He believes that there is a fifty-fifty chance that he would have come across this keepsake if Desdemona was conducting an affair with Cassio, but just a one in ten chance he would have come across it if she was not.

On the basis of these estimates, what is the probability that Othello should assign to Desdemona being unfaithful to him? A reduction in which subjective probabilities would reduce the overall probability that Othello would assign to Desdemona being unfaithful to him?

**References and Links**