The Semimartingale Dynamics and Generator of a Continuous Time Semi-Markov Chain
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Bibliographic record
Abstract
We consider a finite state, continuous time homogeneous semi-Markov chain X = {Xt, t 0}.Without loss of generality the state space of the chain can be identified with the set of unit vectors S = {e 1 , e 2 , . . ., e N } where e i = (0, . . ., 0, 1, 0, . . ., 0) R N .The probabilistic and dynamic properties of X can be described by either a rate matrix A or a matrix which gives the occupation times in the various states together with the probabilities of jumping to a different state.For a continuous time Markov chain the occupation times are memoryless, implying the distributions are exponential.For semi-Markov chains the occupation times can have more general distributions.The relation between these two descriptions is first investigated and the semimartingale dynamics of a semi-Markov chain obtained in contrast to the traditional description of a semi-Markov chain in terms of a renewal process.An equation giving the dynamics of the occupation times is derived together with an equation for the density of the conditional occupation time and state.Some approximations for these dynamics are then obtained.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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