A Combinatorial Approach to Nearly Uncoupled Markov Chains
Bibliographic record
Abstract
A discrete-time Markov chain on a state space S is a sequence of random variables\nX = fx0; x1; : : :g that take on values in S. A Markov chain is a model of a system\nwhich changes or evolves over time; the random variable xt is the state of the system\nat time t.\nA subset E S is referred to as an almost invariant aggregate if whenever xt 2 E,\nthen with high probability xt+1 2 E, as well. That is, if there is a small positive value\n such that if xt 2 E then the probability that xt+1 =2 E is less than or equal to ,\nthen E is an almost invariant aggregate. If E is such an aggregate and xt 2 E, then\nthe probability that xt+1; : : : ; xt+s 2 E is at least (1-E)s. A Markov chain tends to\nremain within its almost invariant aggregates (if it possesses any) for long periods of\ntime.\nWe refer to the Markov chain X as nearly uncoupled (with respect to some positive\n) if its associated state space contains two or more disjoint almost invariant\naggregates. Nearly uncoupled Markov chains are characterised by long periods of\nrelatively constant behaviour, punctuated by occasional drastic changes in state.\nWe present a series of algorithms intended to construct almost invariant aggregates\nof a given Markov chain. These algorithms are iterative processes which utilise a\nconcept known as the stochastic complement. The stochastic complement is a method\nby which a Markov chain on a state space S can be reduced to a random process on\na proper subset S0 S, while preserving many of the algebraic properties of the\noriginal Markov chain.\nWe pay special attention to the reversible case. A Markov chain is reversible if\nit is symmetric in time { by which we mean that if we were to reverse the order of\nthe variables x1; : : : ; xt, for some relatively large t, the resulting process would be\nessentially indistinguishable from the original Markov chain.
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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 | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.010 | 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".