Savile: remember Bayes theorem?

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?

Comments

[Jason]

35.7% is a 'small chance'? I agree with your overall point but that bothered me.

[R. Soul]

I tell ya, all this stuff about Savile nearly ruined my future. I had just gone into production of Jimmy Savile calendars when this story broke. Now the bank is refusing me the funds to launch my product. The project is no longer feasible and I have lost a lot of money.

Still, being the resilient entrepreneur I am, I intend to produce and sell Freddie Starr calendars instead.

[chris]

@ Jason - 35.7% = probably not, and is massively short of the reasonable doubt threshold for conviction.
If a firm sacked someone coz they thought there was a 35.7% chance of them being a criminal, I think there'd often be an outcry.
That's why I called it small.

[Wellington Frink and Hemmersley]

I would imagine that the `incuriosity' stemmed from the fact that Savile had many powerful friends in high places and there is nobody more cowardly than a civil servant. Savile was at the forefront of Thatcher's Big Society forerunner project `Care in the Community'.

Anyway: if you think you may have been molested by Jimmy Savile we can help you claim any compensation due. Text `Howsaboutthathten' to 0799944444 straight away.

[Metatone]

I find it fascinating that media outlets like the Mail and Sun can attack the BBC for not being brave about "what everyone suspected about Jimmy Savile" when they themselves were not brave.

All their fears of libel actions were paralleled by problems for the BBC - Savile moving to ITV, suing the BBC for slander and wrongful dismissal.

Now someone could have gone to the police - but that's really where the Bayesian (unconscious or otherwise) kicks in...

[twitter.com/matt_heath]

35.7% = worth putting some resources into getting more data and making sure kids are properly chaperoned. If that ever was the probability they had in their heads the BBC bosses would certainly have been grossly negligent.

[Account Deleted]

"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?"

Careful. That's someone's sister (or mother) you're talking about there.

[Iain Coleman]

Would you convict someone in a court of law on a 35.7% probability? No.

Would you sack someone on a 35.7% probability? Probably not.

Would you thoroughly investigate serious criminal allegations on a 35.7% probability, and take precautionary actions while the investigation progressed? Damn right.

[Tim Newman]

"Savile was at the forefront of Thatcher's Big Society forerunner project `Care in the Community'."

I did wonder how long it would be before the blame was laid at Thatcher's feet...

[Richard Smith]

Hang on - what's an "allegation"? One specific claim, or lots of different ones made by different people at different times? Does that make a difference?

[chris]

@ Richard. Good question. A lot depends on the correlation between the allegations. If this is low (eg women making allegations without knowing others' are doing so), then the false positive rate will be low. If the rumours are gossip, are the allegations are due to bandwagon-hopping, then we might reasonably ascribe a higher false positive rate to them.

[Howarthm]

Excellent (albeit long) piece on BBC culture from @LRB http://bit.ly/VQkBr7. I'd argue we're still utterly confused about sex and sexuality but that exposure around Savile can only help towards more honest debate.

[rogerh]

Roll back a few years then imagine the screaming and raving from the more hysterical newspapers as the CPS attempts to charge/try Saville on the basis of ancient accusations. If the case ever got near a court imagine the ugly sight of accusers being torn to shreds by well paid lawyers. Bayes theory has been chucked out of UK courts before - it has zero credibility except to statisticians.

A useful line of enquiry might be trawling the files of Saville's lawyers, doubt it will happen tho. I agree with Howarthm, we have barely scratched the surface regarding honest debate about sex, sexuality and human interactions generally.

That was then, processes improved blah blah blah.

[BenSix]

Don't link to Brendan O'Neill. It only encourages him.

[Michael Power]

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

I am afraid you are right about missing the point.

"Wilful blindness" is a problem with cognitive biases, not statistics as Margaret Heffernan explains so clearly in her blogs and book:

http://www.huffingtonpost.co.uk/margaret-heffernan-/jimmy-savile-epidemic-of-wilful-blindness_b_2044966.html

There is a typographical error in the title of the linked blog - it should be "endemic", not "epidemic" wilful blindness as the underlying biases are in all of us.