Measure valued differentiation for stochastic processes : the finite horizon case
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Bibliographic record
Abstract
This paper addresses the problem of sensitivity analysis for finite horizon performance measures of general Markov chains.We derive closed form expressions and associated unbiased gradient estimators for derivatives of finite products of Markov kernels by measure-valued differentiation (MVD).In the MVD setting, derivatives of Markov kernels, called D-derivatives, are defined with respect to an appropriately defined class of performance functions D, such that for any performance measure g D the derivative of the integral of g with respect to the one step transition probability of the Markov chain exists.The MVD approach (1) yields results that that can be applied to performance functions out of a predefined class, (2) allows for a product rule of differentiation, that is, analyzing the derivative of the transition kernel immediately yields finite horizon results, (3) provides an operator language approach to differentiation of Markov chains and (4) clearly identifies the trade-off between the generality of performance classes that can be analyzed and the generality of the classes of measures (Markov kernels).The D-derivative of a measure can be interpreted in terms of various (unbiased) gradient estimators and the product rule for D-differentiation yields a product-rule for various gradient estimators.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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