You may not know the name Abraham Wald, but he has a very valuable lesson you can apply to problem solving, engineering, and many other parts of life. Wald worked for the Statistical Research Group (SRG) during World War II. This was part of a top secret organization in the United States that applied elite mathematical talent to help the allies win the war. Near Columbia University, mathematicians and computers — the human kind — worked on problems ranging from how to keep an enemy plane under fire longer to optimal bombing patterns.
One of Wald’s ways to approach problem was to look beyond the data in front of him. He was looking for things that weren’t there, using their absence as an additional data point. It is easy to critique things that are present but incorrect. It is harder to see things that are missing. But the end results of this technique were profound and present an object lesson we can still draw from today.
Are You Measuring the Wrong Samples?
A problem was posed to the SRG — too many planes weren’t returning from missions. War planes are, of course, the product of engineering trades. Given a certain engine, you can fly so much weight. You divide that weight among crew, equipment, armor, and ordnance. The more you add of one thing, the more you take away from other things. The problem was that Allied planes were being shot down and needed more armor. But more armor meant fewer bombs. The Statistical Research Group’s task was to figure out how to best protect the planes without unnecessarily impacting the other factors.
The Army Air Corps noticed that after a mission, bullet holes were not distributed uniformly across the aircraft. The fuselage took nearly 2 bullet holes per square foot on average, and the fuel system took almost as much. But the engines took just a bit more than one bullet per square foot. The Army wanted to know how much more armor to put on the parts of the plane that were taking the most bullets.
Wald had a different point of view. Instead of putting armor on the fuselage, Wald wanted to add armor to the engines, even though they appeared to be taking fewer hits. Why? Because the samples the Army measured were from planes that returned. Wald surmised that the planes with many bullet holes to the engines were not coming home. The extra armor belonged not on the part of the plane that could survive a lot of bullets, but to the part of the plane that couldn’t.
As a result, extra engine armor appeared on warplanes from that point forward. Once you understand the logic, Wald’s insight seems obvious. But it defies many people’s idea of common sense to protect the part of the plane taking the least damage.
In science and engineering, you have to question your assumptions and the assumptions of others. Things that seem obvious are often wrong. Heavy objects don’t fall faster (with all other things being equal). The Earth isn’t flat despite casual observation from the ground. Continents really do move, but not on a time scale we notice.
Identify Your Assumptions
How often do assumptions bite us when working on hardware or software? All the time. The second hardest bugs to fix are the ones where we’ve made a wrong assumption. For example, you feel sure an input pin is pulled up internally, but it turns out it isn’t. Perhaps you are sure the compiler zeros variables for you, but it turns out it doesn’t. The hardest false assumptions to spot, by the way, are the ones where our abstractions are broken. Finding when a compiler generates incorrect machine code, for example, is very hard. That’s because we’re looking at it from a different level, and everything up there looks just fine. The same happens when you get a black market IC that performs in a similar way, but doesn’t quite meet the specs of a real part.
For software, hardware, and most other fields, it is much easier to look at what’s present and critique it than it is to decide what’s totally missing. In Wald’s case, everyone was looking at the hole densities without thinking why the density wasn’t uniform to start with
Abraham Wald’s Legacy
During his time with the SRG, Wald was not allowed to see the finished reports he contributed to. He was considered an enemy alien, having emigrated to the US from Austria during the war. He was the grandson of a rabbi, the son of a kosher baker, and fled Austria after its annexation by Nazi Germany. Despite not being able to acquire the security clearance, he was content to apply his considerable math expertise to help drive the Nazis out of Europe.
Ironically, Wald and his wife died in a plane crash on an Air India flight. They were survived by their son Robert Wald who went on to become an accomplished theoretical physicist.
There is some debate regarding the veracity of the story of Wald and the missing bullet holes. According to the American Mathematical Society, some parts of it are unverifiable. Wald did work for the SRG and is known for some fundamental techniques of operational research, and it is known that the team worked on aircraft vulnerability. Several people familiar with the work have credited Wald, so I like to think the story is mostly true.
Wald may not have invented this technique, but his application of it was classic, and a lesson from which we can all learn.
26 thoughts on “Abraham Wald’s Problem Solving Lesson Is To Seek What’s Not There”
I’ll just leave this here.
Thanks. I would not have found that wiki and it was very interesting.
Yes, this was critically missing from the article. It’s actually the same in terms of what defines success: don’t jut look at the successful people. Look at those who weren’t successful to understand what results in success and what is just something that everybody does anyways or just isn’t harmful to success.
What do you mean the Earth is not flat? Don’t you know that one third of the world’s population in 2019 believes that? Of course is not something that can be achieved by casual observation from the ground. These people assume what is not there… Wait! What?
This reminds of when fellow named John Craven used Bayes’ Theorem to locate one of our missing submarines (USS Scorpion) that sank when everyone else’s efforts failed. Once he took into account ocean currents, water temperatures at given depths, and weather conditions at the last know location of the sub, etc., it was found right where he said it would be.
Wow, true, but inaccurate in almost every respect. Amazing.
My Scoutmaster in the 1960’s (Robert Eby) had been an air corp colonel and flew P47’s out of Duxford, England – 105 missions! He related to me one day a story about a man he meant who was marking battle damage on outlines of planes when they returned. Bob looked at the drawings and noted to the fellow the puzzling regions where there were not hits. They guy replied that those are the places you get hit if you don’t return. That idea stuck with me. WWII produced such a vast amount of smart clever creativity in such a short time that it boggles the mind. (I transferred all his gun films to video about 2000).
Punctuated equilibrium applied to a society.
“Perhaps you are sure the compiler zeros variables for you and, but it turns out it doesn’t”
Perhaps you edited that sentence’s and, and you no longer needed the and… and…
Inferring the problem from the data that’s missing is a classic solution for parents. No noise? They’re up to something! Something omitted in an account? They’re hiding something.
I’ve used this technique to spot things – low representation of a browser/device in web analytics suggested an issue with loading the site on that device.
“Wald surmised that the planes with many bullet holes to the engines were not coming home. The extra armor belonged not on the part of the plane that could survive a lot of bullets, but to the part of the plane that couldn’t.” But did they analyze aircraft wrecks (when they could be found) to see if the theory matched the facts?
>But did they analyze aircraft wrecks (when they could be found) to see if the theory matched the facts?
In fairness, that would probably lead to a different kind of survivor bias: aircraft which managed to limp at least part of the way home before crashing, and so were more accessible for examination. The wrecks you’d really want to see would the ones which crashed immediately after being shot, which would likely have been strewn around the bombing target or in areas where the enemy had control of the skies, i.e. firmly in enemy territory.
It’s harder to count holes in a heap of twisted metal and strewn bits. Also kind of tricky to go see them in Germany and occupied France.
I meant did they analyze aircraft wreckage after the war, of course you can’t analyze wreckage during the war (having more important stuff to do), but I wondered if they did wreckage analysis after the war, on wreckage they could find, to see if it supported the supposition
Not really. That’s what modern crash analysis teams do. E.g., engine fan blade fractured, shrapnel impacted rotor, decomposition cascaded, fuselage punctured, unscheduled rapid landing. The final crash of the Concorde is a good example of what can be derived from scattered bits of metal.
Did more planes who had the new armor based on his calculations come home consistently? You don’t need wreckage to tell you that.
I keep telling my kids that when they are looking at information, there is *ALWAYS* a filter. I also tell them that they need to make an educated guess as to what that filter is.
doesn’t seem like any fancy math should be required for deciding to put armor on the vulnerable parts of a plane.
You missed the point of the article. The thing you state as obvious was stated as the main task of this research group. The insight was in determining which parts were vulnerable.
Better to read the Wikipedia article.
Survivorship bias: You’re not sampling the population of planes that went out on a mission; you’re examining only those that returned. This was Wald’s observation.
My six year old daughter found in seconds you have to protect the engines …….no big issue
I posted E = MC^2 in seconds. No idea why it was such a feat to come up with it. I can even just Google it!
I knew what the article was about just from reading the title. It was covered years ago in a Car Talk Puzzler.
Apparently this story is part internet myth?
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