The Monday morning quarterbacks are busy today. The Observer accuses the BBC of "institutional incuriosity" about Jimmy Savile and Chris Patten asks:

Can it really be the case that no one knew what he was doing?
Did some turn a blind eye to criminality? Did some prefer not to follow up their suspicions because of this criminal's popularity and place in the schedules? Were reports of criminality put aside or buried?

This misses something - Bayes theorem. This allows us to quantify the probability that someone will be a pervert, in light of an allegation against them. To do so, we need three numbers:

- What is the chance of someone being a pervert, in the absence of evidence against them? This is our prior probability.

- What is the chance of rumours against them, if they were a pervert? This is the chance of a true positive.

- What is the chance of such rumours if they are innocent? This is the false positive rate.

The maths is well explained here. Let's stick some numbers onto this. Let's say we think there's a 10% chance of someone being a pervert, in the absence of evidence. Let's say that, if he were a pervert there's a 100% chance of talk about him; no fire without smoke. And let's say that if he were innocent there's a 20% chance of there being rumours about him. Bayes theorem then tells us that there is a 35.7% chance of him being guilty.

My numbers are arbitary. But they show the point - that even if we give significant credence to the allegations - only a 20% chance of them being false - we should still think there's only a small chance of the accused being a pervert. This is because if there is a good chance someone is innocent, the chance of him being falsely accused is greater than the chance of him being guilty.

Now, I'm not saying BBC bosses consiously applied Bayes theorem; I suspect they were as ignorant of it as their accusers are today. What is possible, though, is that they applied it intuitively, believing that the prior probability of Savile being a pervert was so low that it survived contact with the doubtful allegations against him.

A belief in Savile's probable innocence need not therefore have arisen from turning a blind eye or from incuriosity. It might instead have been the result of maths and logic.

What is possible, though, is that the inputs into this logical process arose from irrational processes. BBC bosses might have attached an unduly low prior probability to Savile being a pervert because of wishful thinking or reframing: "he's not a pervert, just eccentric." And they might have exaggerated the false positive rate, thinking they came from silly girls.

With hindsight, this seems likely. But is it really reasonable to condemn people with hindsight? In the 70s, there was less alertness (panic?) about paedophiles, which generated a lower prior probability then than we'd have now. And would you really have given great credence to the word of the sort of people who screamed and fainted at the sight of the Bay City Rollers?

Yes, it now seems as if BBC bosses were wrong. But Bayes theorem tells us that, given the paucity of hard evidence against Savile in the 70s and 80s, they might not have been as culpably irrational as they seem.

But then, I've missed the point, haven't I?