GPT takes on the Tank Problem
Imagine being handed 10 random page numbers from a book and tasked with estimating the book's total number of pages. Or, suppose you're given 20 random serial numbers from a collection of currency notes and asked to determine the stack's size. Such scenarios capture the essence of a unique challenge faced by World War II statisticians: estimating the total number of German tanks based on the serial numbers of a few captured or destroyed in battle.
This task exemplifies the application of statistical reasoning to solve real-world problems.
Original statistical model
The method adopted by statisticians to solve the German tank problem was quite ingenious. They assumed the serial numbers on the tanks were sequentially issued from a low starting point. By analyzing the serial numbers from the tanks captured or destroyed, they could infer the overall production volume.
Considering a sample of serial numbers like 26, 65, 78, 90, 125, 178, the team tested various strategies, such as using the mean or median of these numbers and then applying a multiplier for estimation.
At the heart of their approach was the use of observed serial numbers to deduce the maximum serial number and, consequently, the total tank count. The simplest estimator used was the maximum-likelihood estimator for the maximum of a uniform distribution:
Estimate?of?total =(sample?size+1) / sample?size)×max of sample
For our example, this would translate to 7/6 × 178=208
The Outcome
The precision of the statistical estimate, compared to other contemporary methods like intelligence reports, is what makes this story captivating. For instance, while statistical methods estimated German tank production at about 246 units in a certain month, other estimates were significantly higher. Post-war records showed the actual production for that month was 245 tanks, underscoring the remarkable accuracy of the statistical estimate and the importance of innovative data interpretation.
How GPT did?
When I threw the Tank problem at ChatGPT with some sample numbers, it quickly shot back with the usual wartime method answer. But I was curious for more, so I asked it to take another crack at figuring out the total number of tanks, this time starting from scratch.
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I nudged GPT to think like a stats whiz and look at the problem from all angles. And yes, it did deliver!
First off, GPT dove into the numbers, finding the average difference between each and adding that to the highest number we had. This move wasn't just new; it showed GPT could think outside the box, offering a new take that made the usual methods look a bit old-hat.
But that wasn't all. I asked for more ideas, and GPT didn't disappoint. It came up with some smart strategies using Bayesian guesswork, clustering algorithms, and even running simulations. Each idea was laid out so clearly and cleverly, really showing off how GPT can handle stats principles in fun and imaginative ways.
Going through this with GPT was like a deep dive into what makes statistical analysis exciting. GPT's knack for switching between different smart solutions, explaining complex stuff in simple ways, and thinking creatively was a real eye-opener to the cool stuff AI can do when tackling tricky problems.
Conclusion
By revisiting the solution to the German tank problem with GPT-4 and first principles, we highlight AI's versatility in not just leveraging classical statistical methods but capacity for innovative problem-solving. This exercise showcases AI's ability to not only grasp and apply historical methodologies but also to explore new strategies based on the underlying principles that guided the original statisticians.
PS : Thank you for reading! The views expressed here are my own, and I invite you to share your thoughts or experiences with AI in problem-solving.
Additional reading :
2. Prompt for picture : Draw a minimalist picture using pencil sketch showing statistical sleuths.