On the Bayesianity of maximum likelihood estimators of restricted location parameters under absolute value error loss
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Bibliographic record
Abstract
We investigate the potential Bayesianity of maximum likelihood estimators (MLE), under absolute value error loss, for estimating the location parameter θ of symmetric and unimodal density functions in the presence of (i) a lower (or upper) bounded constraint, and (ii) an interval constraint, for θ. With these problems being expressed in terms of integral equations, we establish for logconcave densities: the generalized Bayesianity of the MLE in (i); and the proper Bayesianity and admissibility of the MLE in (ii) which extends the normal model result of Iwasa and Moritani. In (i), a key feature concerns a correspondence with a Riemann–Hilbert problem, while in (ii) we use Fredholm´s technique and a contraction mapping argument. We demonstrate that logconcavity is a critical condition with sufficient conditions for non-Bayesianity and, accordingly, with a class of counterexamples. Note that the Bayesianity of the MLE under absolute value loss in the restricted location parameter case is in marked counterdistinction to that under quadratic loss, where, typically, a generalized Bayes estimator must be a smooth function. Finally, various other remarks, illustrations and numerical evaluations are provided.
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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 | 0.027 |
| 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 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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