Bayesian Model Selection for Discrete Graphical Models
Bibliographic record
Abstract
Graphical models allow for easy interpretation and representation of complex distributions. There is an expanding interest in model selection problems for high-dimensional graphical models, particularly when the number of variables increases with the sample size. A popular model selection tool is the Bayes factor, which compares the posterior probabilities of two competing models. Consider data given in the form of a contingency table where N objects are classified according to q random variables, where the conditional independence structure of these random variables are represented by a discrete graphical model G. We assume the cell counts follow a multinomial distribution with a hyper Dirichlet prior distribution imposed on the cell probability parameters. Then we can write the Bayes factor as a product of gamma functions indexed by the cliques and separators of G.\n\nIn this thesis, we study the behaviour of the Bayes factor when the dimension of a true discrete graphical model is fixed and when the dimension increases to infinity with the sample size. We prove that the Bayes factor is strong model selection consistent for both decomposable and non-decomposable discrete graphical models. When the true graph is non-decomposable, we prove that the Bayes factor selects a minimal triangulation of the true graph. We support our theoretical results with various simulations. \n\nIn addition, we introduce a variation of the genetic algorithm, called the graphical local genetic algorithm, which can be implemented on large data sets. We use a local search operator and a normalizing constant proportionate to the posterior probability of the candidate models to determine optimal submodels, then reconstruct the full graph from the resulting subgraphs. We demonstrate the graphical local genetic algorithm's capabilities on both simulated data sets with known true graphs and on a real-world data set.
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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 | 0.014 | 0.053 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 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".