Algorithmic barriers to representing conditional independence
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Bibliographic record
Abstract
We define a represention of conditional independence in terms of products of probability kernels, and ask when such representations are computable. We pursue this question in the context of exchangeable sequences and arrays of random variables, which arise in statistical contexts. Exchangeable sequences are conditionally i.i.d. by de Finetti's theorem. Known results about the computability of de Finetti's theorem imply that these conditional independences are computable. The conditional independences underlying exchangeable arrays are characterized by the Aldous-Hoover theorem. In the special case of adjacency matrices of undirected graphs, i.e., symmetric binary arrays, this representation theorem expresses the conditional independences in terms of graphons. We prove that there exist exchangeable random graphs that can be computably sampled but whose corresponding graphons are not computable as functions or even as L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> equivalence classes. We also give results on the approximability of graphons in certain special cases.
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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 | 0.000 |
| 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 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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