In 1957, at the height of the Cold War, statistician Jimmie Savage of the RAND institute was asked a scarily-intriguing question: ‘What is the probability that one of our own thermonuclear bombs might explode by accident?’
The Strategic Air Command was in fact blasé about this possibility. In the twelve years since Hiroshima and Nagasaki there had been plenty of nuclear tests but no accidental detonations. So by definition, the probability of a catastrophe had to be zero, or pretty close.
Jimmie, however, knew better, because he had access to classified files detailing a number of chilling nuclear accidents. In 1950 two nuclear-armed USAF bombers had crashed with detonation of their bombs’ conventional explosives: thankfully the nuclear cores had not followed suit. In other cases hydrogen bombs had fallen out of insecure bomb-bays onto American soil, or had been jettisoned into the ocean.
Savage turned to Bayes’ Theorem, a method for adjusting prior beliefs based on new evidence. He interviewed weapons scientists and air force staff to determine all the possible ways an accidental thermonuclear detonation could occur to create a Bayesian model of nuclear disaster. He then had to put in the numbers, but since these are classified, we will illustrate Jimmie’s thinking based on a plausible model.
Call the existing United States Air Force’s estimate of the probability of an accidental nuclear detonation p(D). The Strategic Air Command didn’t really believe that the chances of a nuclear explosion were zero, they just thought they were negligible: say one in a thousand per year. So we’ll put p(D) = 0.001.
Now consider the probability of an accident, a critical weapons event, p(E). The experts had itemised the events which could lead to such incidents and in addition, Jimmie had his alarming case file. It seemed that the probability of a critical weapons event would be at least 5% per year.
What we need is the revised probability of an accidental nuclear explosion, based on Jimmie’s expert analysis, which we’ll call p(D’).
Finally, we need to estimate p(E|D) – the probability that if a detonation were to happen, a previous critical weapons event would have been the cause. This is almost certain so we’ll call it 1.
Now we can apply Bayes’ Theorem.
p(D’) = p(D) * p(E|D)/p(E).
Putting in the numbers:
p(D’) = 0.001 * 1/0.05 = 2%.
What is this telling us? That given a hard-headed look, the chances of the USAF experiencing an accidental thermonuclear explosion over a year were not one in a thousand, but almost twenty times bigger, more like two percent.*
Savage thought he would be mincemeat when he presented his findings to tough, cigar-chomping Air Force General Curtis LeMay. He was, however, to be surprised.
LeMay in fact moved fast to issue orders for new safeguards – coded locks on the nukes and use of the ‘two man rule’. Lending urgency, it turned out that even before the new procedures could be fully implemented, an SAC B52 bomber disintegrated in flight over North Carolina. One of its 24 megaton bombs fell into a swamp where it still resides today – but only one of the six pre-existing safety devices worked.
Bayes’ theorem is controversial when used to update prior beliefs. Where did that one-in-a-thousand prior probability p(D) really come from? After all, there had never been an accidental detonation. But often the exact prior doesn’t really matter; it’s swamped by the value of new information based on new analysis. Bayesian analysis can force people out of their comfort zone, and sometimes that’s really important.
The example above was based on Chapter 9 of Sharon Bertsch McGrayne’s new book: “The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy” (Yale University Press, 2011). Sharon tells a riveting story of eccentric intellectuals, academic conflict and strange but vital applications, in a page-turning adventure.
If you wish to learn any of the relevant mathematics, however, you will be wasting your time. There is no math at all in the main text, and in the appendix where Sharon puts some numbers in the formula to show how it might work, amazingly she gets it wrong (she doesn’t understand conditional probability).
The book was recommended by both The Sunday Times and The Economist – I guess for light reading. The Wikipedia articles here and here are useful if you want to really find out how Bayesian inference actually works.
* You may notice that the smaller p(E) is, the more likely it is that an accidental detonation will take place. Smaller chances of accidents lead to higher chances of nuclear explosions? Why is that?
Recall that p(D’) – really p(D|E) – is the probability of a detonation given that the critical weapons event has already occurred. We already know that the probability for D is low and that D can’t happen unless E has happened first.
So as p(E) gets smaller, the chances of E resulting in D rather than leading to an alternative and relatively harmless outcome just increase. Remember, p(D’) is the probability of a detonation given that E has already occurred.
The logic of the prior SAC position was that yes, accidents do happen quite frequently but they will virtually never lead to a catastrophic nuclear accident – so large p(E), low p(D). What Jimmie showed was that an important category of accidents was indeed exceedingly dangerous.