Estimation Efficiency Under Privacy Constraints
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Bibliographic record
Abstract
We investigate the problem of estimating a random variable Y under a privacy constraint dictated by another correlated random variable X. When X and Y are discrete, we express the underlying privacy-utility tradeoff in terms of the privacy-constrained guessing probability (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">XY</sub> , ε), and the maximum probability Pc(Y|Z) of correctly guessing Y given an auxiliary random variable Z, where the maximization is taken over all P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z|Y</sub> ensuring that Pc(X|Z) ≤ ε for a given privacy threshold ε ≥ 0. We prove that ħ (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">XY</sub> , ·) is concave and piecewise linear, which allows us to derive its expression in closed form for any ε when X and Y are binary. In the non-binary case, we derive (P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">XY</sub> , ε) in the high-utility regime (i.e., for sufficiently large, but nontrivial, values of ε) under the assumption that Y and Z have the same alphabets. We also analyze the privacy-constrained guessing probability for two scenarios in which X, Y, and Z are binary vectors. When X and Y are continuous random variables, we formulate the corresponding privacy-utility tradeoff in terms of sENSR(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">XY</sub> , ε), the smallest normalized minimum mean squared-error (mmse) incurred in estimating Y from a Gaussian perturbation Z. Here, the minimization is taken over a family of Gaussian perturbations Z for which the mmse of f (X) given Z is within a factor 1-ε from the variance of f (X) for any non-constant real-valued function f . We derive tight upper and lower bounds for sENSR when Y is Gaussian. For general absolutely continuous random variables, we obtain a tight lower bound for sENSR(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">XY</sub> , ε) in the high privacy regime, i.e., for small ε.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.004 |
| Open science | 0.007 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.002 |
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 it