Monte Carlo and Quasi Monte Carlo Approach to Ulam's Method for Position Dependent Random Maps
Bibliographic record
Abstract
We consider position random maps $T=\{\tau_1(x),\tau_2(x),\ldots, \tau_K(x); p_1(x),p_2(x),\ldots,p_K(x)\}$ on $I=[0, 1],$ where $\tau_k, k=1, 2, \dots, K$ is non-singular map on $[0,1]$ into $[0, 1]$ and $\{p_1(x),p_2(x),\ldots,p_K(x)\}$ is a set of position dependent probabilities on $[0, 1]$. We assume that the random map $T$ posses a density function $f^*$ of the unique absolutely continuous invariant measure (acim) $\mu^*$. In this paper, first, we present a general numerical algorithm for the approximation of the density function $f^*.$ Moreover, we show that Ulam's method is a special case of the general method. Finally, we describe a Monte-Carlo and a Quasi Monte Carlo implementations of Ulam's method for the approximation of $f^*$. The main advantage of these methods is that we do not need to find the inverse images of subsets under the transformations of the random map $T$.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".