Bibliographic record
Abstract
The German Tank Problem dates back to World War II when the Allies used a statistical approach to estimate the number of enemy tanks produced or on the field from observed serial numbers after battles. Assuming that the tanks are labeled consecutively starting from 1, if we observe k tanks from a total of N tanks with the maximum observed tank being m, then the best estimate for N is m(1 + 1/k) - 1. We refer to an estimate as "best" when the estimate is closest to the actual number of tanks. We explore many generalizations; first, we looked at the discrete and continuous one-dimensional case. We attempted to improve the original formula by using different estimators such as the second largest and Lth largest tank, and applied motivation from portfolio theory by seeing if a weighted average of different estimators would produce less variance; however, the original formula, using the largest tank proved to be the best; the continuous case was similar. Then, we looked at the discrete and continuous square and circle variants where we pick pairs instead of points, which were more complex as we dealt with problems in geometry and number theory, such as dealing with curvature issues in the circle, and the problem that not every number is representable as a sum of two squares. In some cases, when we could not derive precise formulas, we derived approximate formulas. For the discrete and continuous square, we tested various statistics, but found that the largest observed component of our pairs is the best statistic to look at; the scaling factor for both cases is (2k+1)/2k. For the circle we used motivation from the equation of a circle; for the continuous case, we looked at the square root of X2+Y2 and for the discrete case, we looked at X2+Y2 and took a square root at the end to estimate for r. Interestingly, the scaling factors, a number, generally a little greater than 1, that we multiplied to scale up to get our estimation, were different for the cases. Lastly, we generalized the problem into L-dimensional squares and circles. The discrete and continuous square proved to be similar to the two-dimensional square problem. However, for the Lth dimensional circle, we had to use formulas for the volume of the L-ball, and had to approximate the number of lattice points inside it. The discrete circle formula was particularly interesting, as there was no L dependence in the formula.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.025 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.024 | 0.003 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".