This blog’s very first post dealt with how simple mathematics was used to fight Nazis in World War II, and that has since become one of favorite applications of math. While reading about the war, I recently came across the German Tank Problem, and I found it even more fascinating than that case with Abraham Wald (especially because it involves more Bayesian Theory). One of the critical aspects of WWII was intelligence, which provided the basis for a country’s actions, and the Allies invested plenty of resources into trying to estimate the extent of German forces, and especially tanks. Captured or destroyed tanks always had a serial number, and therefore the number of tanks could be estimated through this data. Statisticians had been doing this for ages, but what made this case different was the employment of Bayesian inference rather than Frequentist inference to obtain much more accurate results. I saw a great explanation of Frequentist vs. Bayesian thinking on stackexchange:
Frequentist Reasoning — I can hear the phone beeping. I also have a mental model which helps me identify the area from which the sound is coming. Therefore, upon hearing the beep, I infer the area of my home I must search to locate the phone.
Bayesian Reasoning — I can hear the phone beeping. Now, apart from a mental model which helps me identify the area from which the sound is coming from, I also know the locations where I have misplaced the phone in the past. So, I combine my inferences using the beeps and my prior information about the locations I have misplaced the phone in the past to identify an area I must search to locate the phone.
The Bayesian reasoning allowed for a more substantiated estimate of the scale of the tank forces. In this way, the values of the serial numbers was used to ascertain the probability that the total number of tanks was a certain n, and then all these probabilities were compiled. The mathematics of this turns out to be very interesting, and I’m just going to outline it briefly.
Let us take the example of the 4 serial numbers of 19, 40, 42, and 60. We know that the highest number is 60, and therefore can find the probability that this is the highest number for different numbers of tanks. Given that there are 100 tanks, the probability that the highest number is 60 in 4 samples is the total number of possibilities with 60 as the highest divided by total possibilities, which is 59C3/100C4. This is because one sample has to be the maximum, and all the others have to be lower than that so our possibilities are limited to 59C3. This generalizes to:
The variable n is the number of tanks, m is the maximum serial number, and k is the number of samples. Now, the expected number of tanks is simply the expected value of the number of tanks when the probability is multiplied by n and added for all possible values of n, which is anything above m, the maximum. This can be summarized by the following:
This can be algebraically proven to equal:
This generalization of the series can be arrived at using some algebra and the following identity:
Now that we have the value for the mean, it is easy to find the standard deviation using the fact that the std. deviation is sq. root of the variance, where variance is:
Now this becomes an algebraic exercise, as we solve for σ where variance is σ2.
Plugging in the values of m, n, and k for the set of data of 19, 40, 42, and 60, we come to:
This estimate has a considerable uncertainty, however is objectively better than that of wartime intelligence agencies, which often came to wild numbers such as 1400. Yet again, we can see that just brute force didn’t win the war, but a steady supply of some of the Allies’ brightest minds to apply their mathematics ingeniously.