Uncertainty, entropy, variance and the effect of partial information
Bibliographic record
Abstract
Uncertainty about the value of an unmeasured real random variable Y is commonly represented by either the entropy or variance of its distribution.If it becomes known that Y lies in a subset A of the support of V's distribution, one might expect uncertainty about Y to decrease.In other words, one might expect the entropy and variance of V's conditional distribution given Y E A to be less than their counterparts for the unconditional distribution.Going further it might be conjectured that the uncertainty about Y would be greater given the knowledge that Y E B as compared with Y G A C B.We do not know whether these conjectures are correct.However, we give sufficient conditions in certain cases where they are true.In particular, when Y is normally distributed we can make considerable progress.For example, we show in the case that A = [α, b] and Y normally distributed with mean η and variance 1, that the variance of the conditional distribution of Y given that α < Y < b is less than that of the unconditional distribution, thereby confirming our intuitive reasoning in this case.This last example also shows that for this exponential family the variance is less than 1 for all α < b and all η-a result that is not known among the experts on exponential families we consulted.The only relevant thing is uncertainty-the extent of our own knowledge and ignorance, de Finetti (1970, Preface, pp.xi-xii)
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.007 | 0.033 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.007 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".