Approximations and Scaling Llimits of Markov Chains with Applications to MCMC and Approximate Inference
Bibliographic record
Abstract
Markov chains are an essential tool in computational statistics because they form the basis for efficient exact and approximate inference methods, especially in Bayesian statistics. This dissertation offers insight into the viability of approximate inference methods based on both approximations to the transition kernels of a Markov chain for exact methods, and on Markov chains derived from unadjusted stochastic gradient methods. This dissertation also demonstrates how to tune computational methods based upon Markov chains in order to optimize efficiency and accuracy for approximate and exact inference. Results are obtained via two key theoretical methods: (1) an analysis of the perturbation sensitivity of Markov chains using operator theory, and (2) through scaling limits of Markov chains that facilitate a comparison to idealized continuous-time processes. The primary contributions of this dissertation are: (i) a perturbation analysis of reversible geometrically ergodic Markov chains, which characterizes the stability of the stationary distribution and rate of convergence under changes in the transition dynamics; (ii) results on the geometry of probability densities, generalized distributional integration-by-parts, and their consequences; (iii) a joint characterization of the optimal proposal scaling and shaping for the random-walk Metropolis algorithm; and (iv) a complete characterization of the statistical asymptotics of stochastic gradient algorithms as methods for approximate inference, with recommendations on how to tune them for accuracy and efficiency.
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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 | 0.006 | 0.041 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 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".