Specifically, during the war the engineers took a Bayesian approach and put armor on the bombers in locations where bombers which came back did not have damage. Reason being that hits in those sections of the planes were more likely than not resulting in fatalities. Smart thinking.

It appears that in the original 1943 memoranda by Wald takes a frequentist approach; see link below.

"A METHOD OF ESTIMATING PLANE VULNERABILITY BASED ON DAMAGE OF SURVIVORS" BY ABRAHAM WALD (1943) : http://www4.ncsu.edu/~swu6/documents/A_Reprint_Plane_Vulnera...

Definitely smart thinning, but not sure what makes it Bayesian.

We're interested in the quantity:

  (1) P(crash | section hit)

and adding armour to those sections where that quantity is maximized (maybe with some thought to the relative weight of armour needed for each section, but I digress).

Let's directly apply Bayes' rule:

  (2) P(crash | section hit) = P(section hit | crash) * P(crash) / P(section hit)

The denominator can be further expanded:

  (3) P(section hit) = P(section hit | crash) * P(crash) + P(section hit | no crash) * P(no crash)

So we can see from the 2nd term that if aircraft regularly comes back with a section that's been hit and yet it hasn't crashed, then that directly reduces (1), meaning that section needs less relatively less protection, all else being equal.

Another point in this method's favor is if crashed aircraft frames are too damaged to permit us to identify which sections were damaged. In that case, we can still estimate (1) just by replacing all the P(section hit | crash) terms with a uniform term.

This analysis can be further expanded to the actual amount of damage each section took in a as well. The more damage a section took on surviving aircraft, the less protection it needs.

anti-aircraft fire is random so doesn't specifically target any section of the plane. Therefore, the planes that return to base are a biased sample of the planes that took on damage. See: http://www4.ncsu.edu/~swu6/documents/A_Reprint_Plane_Vulnera...

Funny thing is, they used all this advanced math to reach the same conclusion the 5 years old child would come to: armor should be put around engine and ammunition storage. Too much education sometimes makes you really stupid.

AA hits planes on the bottom, presumably they weren't adding all the armor to the top?

Does it? AA explodes at predetermined altitudes which may be above or below the plane. The resulting shrapnel....

It's not like a rifle bullet.

There's the shell itself that can still hit a plane, also proximity mines, which I guess would be more likely to go off on the way app (unless there is a delay after the proximity is triggers). The air force would also know the approximate limit of the enemies AA fire and site above it if possible.