Model-Based Smoothing with Integrated Wiener Processes and Overlapping Splines
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Bibliographic record
Abstract
In many applications that involve the inference of an unknown smooth function, the inference of its derivatives is also important.To make joint inferences of the function and its derivatives, a class of Gaussian processes called pth order Integrated Wiener's Process (IWP), is considered.Methods for constructing a finite element (FEM) approximation of an IWP exist but only focus on the case p = 2 and do not allow appropriate inference for derivatives.In this article, we propose an alternative FEM approximation with overlapping splines (O-spline).The O-spline approximation applies for any order p Z + , and provides consistent and efficient inference for all derivatives up to order p -1.It is shown both theoretically and empirically that the O-spline approximation converges to the IWP as the number of knots increases.We further provide a unified and interpretable way to define priors for the smoothing parameter based on the notion of predictive standard deviation, which is invariant to the order p and the knot placement.Finally, we demonstrate the practical use of the O-spline approximation through an analysis of COVID death rates where the inference of derivative has an important interpretation in terms of the course of the pandemic.
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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 | 0.002 |
| 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