Dec 042012

In the unlikely Comedy of Othello, the love struck Moor of Venice may simply refuse to contemplate the possibility of betrayal: BR=0% means PP=0%. Blind Faith is impermeable to all evidence – and wide open to comical consequences. A wiser Othello must therefore concede that there is a minuscule but non zero chance that Desdemona is betraying him. He knows that there is a level of evidence – e.g. seeing her in bed with Cassio – that would leave him no alternative. As long as BR is non zero, a Smoking Gun – conclusive evidence, logically incompatible with the tested hypothesis being false – would force him to acknowledge the bitter truth: FPR=0% means PP=100%.

An even wiser Othello would allow for a more sizeable Base Rate. True, Desdemona is the one and only, but the prior probability that she is an adulterer cannot be left to wishful thinking: it should be grounded on impartial data. But this is easier said than done, and not just for the practical difficulty of finding the numbers, but for the conceptual one of defining the relevant reference class. Should Othello look for the percentage of female adulterers in Venice? This would give him the probability that a randomly selected Venetian woman is an adulterer. But is this what he is after? Surely Desdemona is not an average Venetian. Should he limit the sample to women who are more like Desdemona? But what does that mean? Choosing an appropriate reference class is a fundamental problem of statistics, which renders data much less ‘objective’ than it may appear at first sight. Prior probabilities are themselves conditional on background evidence. Ultimately, the accumulation of evidence leads to the truth. But starting from a 4% Base Rate rather than one-in-a-million makes a lot of difference: with TPR=80% and FPR=1%, the Posterior Probability of infidelity, given the handkerchief evidence, is 77% in the first case and 0.01% in the second. As in the Humean dictum: Extraordinary claims require extraordinary evidence.

Choosing an appropriate Base Rate is such a difficult and ultimately arbitrary decision that one is tempted to admit defeat, declare total ignorance and assume prior indifference: “Look, I have no idea if Desdemona is betraying me or not. She may, she may be not, I simply don’t know.” Once Othello, nudged by the wily Iago, falls prey to the Prior Indifference Fallacy, he’s done: the handkerchief in Cassio’s lodgings gives him virtual certainty of betrayal.

Extreme trust – a minuscule BR – is one way to avoid this mistake. But a much better one, grounded on evidence, not hope, would be TPR=0%. This is the mirror image of the Smoking Gun: just as FPR=0% implies PP=100%, irrespective of BR and TPR, TPR=0% implies PP=0%, irrespective of BR and FPR. Othello would love to find evidence that is logically incompatible with the tested hypothesis being true. The handkerchief won’t do: Desdemona may try to convince Othello that she would not have left it behind in case of betrayal, but a wised-up Othello won’t buy it. He needs evidence that admits no True Positives: a Perfect Alibi. Cassio is gay, for example.

Conclusive evidence – a Smoking Gun or a Perfect Alibi – relieves us from the difficult task of estimating the Base Rate. With conclusive evidence in hand, we can comfortably admit ignorance, profess prior indifference and let the evidence speak for itself. Probability becomes Certainty. Certainty is different from Faith: Faith is BR=0% or 100%; Certainty is FPR=0% or TPR=0%. Sherlock Holmes is the supremo of conclusive evidence: “When you have eliminated the impossible, whatever remains, however improbable, must be the truth” (The Sign of Four, Chapter 6). But outside the world of fiction, certainty is a rare occurrence: in most cases we are left to deal with uncertainty.

Lacking extreme trust and a Perfect Alibi, what else could have saved Desdemona? Othello might have adopted Silver’s spouse assumptions: BR=4%, TPR=50%, FPR=5%, hence PP=29%. Still a big torment for the fiery Moor, but probably low enough to avoid tragedy. If Othello thinks a 5% FPR is too high for the handkerchief evidence – he’s blind to Iago’s scheming and insists on 1% – then Desdemona should keep working on lowering TPR. 10% would do. Good luck.

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