So what am I smoking and barking about?
It is about the vast probability gap between Inspector Hubbard and the Court. The Inspector thinks Margot is innocent. The Court thinks she is guilty. Only one of them is right. To see who, conclusive evidence is desirable. But the best conclusive evidence differs in the two cases. For the Inspector, a Smoking Gun is near perfect evidence: if positive, Margot is certainly guilty; if negative, she is almost certainly innocent. For the Court, a Barking Dog is near perfect evidence: if negative, Margot is certainly innocent; if positive, she is almost certainly guilty.
On the other hand, with a high prior of guilt a Smoking Gun would be imperfect evidence: if positive, Margot is certainly guilty; but if negative, she remains quite likely guilty. And with a low prior of guilt a Barking Dog would also be imperfect evidence: if negative, Margot is certainly innocent; but if positive, she remains quite likely innocent.
Incidentally, a Strangler Tie is the negative equivalent of a Smoking Gun, and a Perfect Alibi is the positive equivalent of a Barking Dog. Otherwise, they work in the same way. Here is the full picture:
As a result, the best way for the Inspector to be sure about Margot is not to prove her innocence, but to try to gain conclusive evidence – positive (Smoking Gun) or negative (Strangler Tie) – that proves her guilty. Vice versa, the best way for the Court to be sure about Margot is not to prove her guilty, but to try to gain conclusive evidence – positive (Perfect Alibi) or negative (Barking Dog) – that proves her innocence. It is the failure to prove her guilt that makes Margot almost certainly innocent in the eyes of the Inspector. It is the failure to prove her innocence that makes Margot almost certainly guilty in the eyes of the Court.
What about Tony? Before the door test, he is to the Inspector what Margot is to the Court: very likely guilty. So why does the Inspector put him through the same Smoking Gun test that he run on Margot? It is not because a Smoking Gun would be a near perfect test in the eyes of the Court. True, the Court decides, and they think Tony is innocent. But before trying to convince the Court, the Inspector needs to make up his own mind. In that respect, a Smoking Gun won’t do: if Tony fails to open the door (as he was about to), the Inspector will still suspect him. He would be better served by evidence aimed at proving Tony’s innocence. Given his priors, a Barking Dog (or a Perfect Alibi) would be near perfect. For instance, he could investigate into whether Tony had recently met Swann: if he didn’t, he would be certainly innocent; if he did, in the eyes of the Inspector he would be almost certainly guilty.
So why does the Inspector still prefer a Smoking Gun? It is because, as a conscientious professional, he adheres to the Blackstone Principle: “It is better that ten guilty persons escape than that one innocent suffer”. Convicting an innocent person – a False Positive – is a much more harmful mistake than acquitting a guilty person – a False Negative. A Smoking Gun completely avoids False Positives: FPR=0% – that is why the Inspector likes it. If Tony is innocent, there is no way that he will open the door. So, if he opens the door, everybody agrees he is certainly guilty. If he doesn’t, the Inspector will still suspect him, but his mind will be at peace thinking that he has run no risk of partaking in the conviction of a possibly innocent person.
But that is a mistake. With a high prior of guilt – say BR=90% – a Barking Dog is near perfect evidence. With FPR=1-TNR=5%, the probability of guilt if Tony met Swann is 99%. And even with FPR=20%, it is 98%. Tony is not certainly guilty, but he is guilty beyond reasonable doubt. That should be enough, even for a conscientious Blackstonian Inspector.
The next post will reveal what I am really smoking and barking about.