Two behavioural pseudometrics for continuous-time Markov processes
Bibliographic record
Abstract
Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in discrete-time settings where a key notion is the bisimulation metric which quantifies "how similar two states are". In [ 11], we generalized the concept of bisimulation metric in order to metrize the behaviour of continuous-time Markov processes. Similarly to the discrete-time case, we constructed a pseudometric following two iterative approaches - through a functional and through a real-valued logic, and showed that the outcomes coincide: the pseudometric obtained from the logic is a specific fixpoint of the functional which yields our first pseudometric. However, different from the discrete-time setting, in which the process has a step-by-step dynamics, the behavioural pseudometric we constructed applies to Markov processes that evolve continuously through time, such as diffusions and jump diffusions. While our treatment of the pseudometric in [11] relied on the time-indexed Markov kernels, in [ 8 , 9, 10 ], we showed the importance of trajectories in the consideration of behavioural equivalences for true continuous-time Markov processes. In this paper, we take the work from [11 ] further and propose a second behavioural pseudometric for diffusions based on trajectories. We conduct a similar study of this pseudometric from both the perspective of a functional and the viewpoint of a real-valued logic. We also compare this pseudometric with the first pseudometric obtained in [11].
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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 | 0.024 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".