QMC integration errors and quasi-asymptotics
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Abstract
Abstract A crude Monte Carlo (MC) method allows to calculate integrals over a d -dimensional cube. As the number N of integration nodes becomes large, the rate of probable error of the MC method decreases as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>O</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:msqrt> <m:mi>N</m:mi> </m:msqrt> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {O(1/\sqrt{N})} . The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of d -dimensional functions contains a multiplier <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {1/N} . However, the multiplier <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ln</m:mi> <m:mo></m:mo> <m:mi>N</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>d</m:mi> </m:msup> </m:math> {(\ln N)^{d}} is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {1/N} . However, our numerical experiments show that using quasi-random points of Sobol sequences with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>=</m:mo> <m:msup> <m:mn>2</m:mn> <m:mi>m</m:mi> </m:msup> </m:mrow> </m:math> {N=2^{m}} with natural m makes the integration error approximately proportional to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> </m:math> {1/N} . In our numerical experiments, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≤</m:mo> <m:mn>15</m:mn> </m:mrow> </m:math> {d\leq 15} , and we used <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>≤</m:mo> <m:msup> <m:mn>2</m:mn> <m:mn>40</m:mn> </m:msup> </m:mrow> </m:math> {N\leq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≤</m:mo> <m:msup> <m:mn>2</m:mn> <m:mn>14</m:mn> </m:msup> </m:mrow> </m:math> {d\leq 2^{14}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>≤</m:mo> <m:msup> <m:mn>2</m:mn> <m:mn>63</m:mn> </m:msup> </m:mrow> </m:math> {N\leq 2^{63}} .
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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