Any sign that can be related to a hypothesis is a form of evidence about the hypothesis. If the sign is there, we say the evidence is positive. If it is not there, we say the evidence is negative.
Evidence can be more or less accurate, with accuracy defined as A=1/2+(TPR-FPR)/2. Perfect evidence is perfectly accurate: A=1, i.e. TPR=1 and FPR=0. Perfect contrary evidence has A=0, i.e. TPR=0 and FPR=1. Perfect accuracy means that the evidence, whether positive or negative, is conclusive. With A=1, if the evidence is positive, the hypothesis must be true; if it is negative, the hypothesis must be false. With A=0, if the evidence is positive, the hypothesis must be false; if it is negative, the hypothesis must be true. Perfect evidence brings perfect certainty: the hypothesis is true if and only if the evidence is positive (A=1) or negative (A=0). Hence we say that the evidence causes the hypothesis to be true (or false).
Conclusive evidence also brings certainty, but it is not perfect. A Smoking Gun is conclusive evidence that the hypothesis must be true. A Perfect Alibi is conclusive evidence that the hypothesis must be false. The first is incompatible with False Positives (FPR=0), the second with True Positives (TPR=0). However, while conclusive if the evidence is positive, Smoking Guns and Perfect Alibis are not conclusive if the evidence is negative.
Let’s see. Bayes’ Theorem for negative evidence is:
NP is the Posterior Probability of the hypothesis, given negative evidence. FNR=1-TPR is the False Negative Rate and TNR=1-FPR is the True Negative Rate.
Perfect evidence is incompatible with False Positives and False Negatives: FPR=0 and FNR=0, hence PP=1 and NP=0. If contrary, it is incompatible with True Positives and True Negatives: TPR=0 and TNR=0, hence PP=0 and NP=1. Perfect evidence, whether positive or negative, is always infallible.
Smoking Guns and Perfect Alibis are infallible if the evidence is positive, but not if the evidence is negative. If there is a Smoking Gun, the hypothesis must be true: PP=1; but if there isn’t one, the hypothesis is not necessarily false: NP>0. If there is a Perfect Alibi, the hypothesis must be false: PP=0; but if there isn’t one, the hypothesis is not necessarily true: NP<1.
I know your head is spinning. So was mine. But it is quite simple. Smoking Guns and Perfect Alibis get their name from the hypothesis of guilt. A Smoking Gun is conclusive evidence that the suspect is guilty. A Perfect Alibi is conclusive evidence that he is innocent. But if no Smoking Gun is found, it does not mean that the suspect is necessarily innocent. And if he doesn’t have a Perfect Alibi, it does not mean that he is necessarily guilty.
Conclusive evidence is often used in works of fiction to bring out final certainty. Sherlock Holmes is the maestro of conclusive evidence. His incessant accumulation of evidence often culminates with a conclusive piece, thanks to which his deductions about guilt or innocence leave the realm of probability and, through inescapable logic, reach the pinnacle of certainty. In many of Alfred Hitchcock’s movies, the main character is an innocent man, being cornered by an accumulation of circumstantial evidence pointing to his guilt, until a single piece of conclusive evidence proves his innocence.
When the great race horse Silver Blaze disappears, and his trainer, John Straker, is found murdered, the main suspect is the bookmaker Fitzroy Simpson. But Sherlock Holmes proves that Simpson could not have killed Stracker, because the dog didn’t bark:
“Is there any point to which you would wish to draw my attention?” “To the curious incident of the dog in the night-time.” “The dog did nothing in the night-time.” “That was the curious incident,” remarked Sherlock Holmes.
In the final scene of Hitchcock’s Frenzy, Inspector Oxford nails down the strangler: “Mr. Rusk, you’re not wearing your tie”, thus proving that Dick Blaney — until then the chief suspect — is innocent.
As these two stories show, negative evidence can be as conclusive as positive evidence.
A Barking Dog is evidence incompatible with False Negatives: FNR=0. From Bayes’ Theorem, NP=0: given negative evidence, the hypothesis must be false.
A Strangler Tie is evidence incompatible with True Negatives: TNR=0. From Bayes’ Theorem, NP=1: given negative evidence, the hypothesis must be true.
Barking Dogs and Strangler Ties are infallible if the evidence is negative, but not if it is positive. Since the dog didn’t bark, Simpson must be innocent: NP=0. Since Rusk was not wearing a tie, he must be guilty: NP=1. But if the dog had barked, Simpson would not have been necessarily guilty: PP<1. If Rusk had been wearing a tie, he would not have been necessarily innocent: PP>0.
Against head spinning, the following table summarizes the four cases:
How imperfect can conclusive evidence be? I’m sure you can’t wait to find out.
Originally published at Bayes.