Bayesian precision and covariance matrix estimation for graphical Gaussian models with edge and vertex symmetries
Bibliographic record
Abstract
Graphical Gaussian models with edge and vertex symmetries were introduced by Højsgaard & Lauritzen (2008), who gave an algorithm for computing the maximum likelihood estimate of the precision matrix for such models. In this paper, we take a Bayesian approach to its estimation. We consider only models with symmetry constraints and which thus form a natural exponential family with the precision matrix as the canonical parameter. We identify the Diaconis–Ylvisaker conjugate prior for these models, develop a scheme to sample from the prior and posterior distributions, and thus obtain estimates of the posterior mean of the precision and covariance matrices. Such a sampling scheme is essential for model selection in coloured graphical Gaussian models. In order to verify the precision of our estimates, we derive an analytic expression for the expected value of the precision matrix when the graph underlying our model is a tree, a complete graph on three vertices, or a decomposable graph on four vertices with various symmetries, and we compare our estimates with the posterior mean of the precision matrix and the expected mean of the coloured graphical Gaussian model, that is, of the covariance matrix. We also verify the accuracy of our estimates on simulated data.
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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 | 0.017 | 0.111 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.004 | 0.003 |
| Research integrity | 0.003 | 0.004 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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".