The Poisson Maximum Entropy Model for Homogeneous Poisson Processes
Bibliographic record
Abstract
Our main interest is parameter estimation using maximum entropy methods in the prediction of future events for Homogeneous Poisson Processes when the distribution governing the distribution of the parameters is unknown. We intend to use empirical Bayes techniques and the maximum entropy principle to model the prior information. This approach has also been motivated by the success of the gamma prior for this problem, since it is well known that the gamma maximizes Shannon entropy under appropriately chosen constraints. However, as an alternative, we propose here to apply one of the often used methods to estimate the parameters of the maximum entropy prior. It consists of moment matching, that is, maximizing the entropy subject to the constraint that the first two moments equal the empirical ones and we obtain the truncated normal distribution (truncated below at the origin) as a solution. We also use maximum likelihood estimation (MLE) methods to estimate the parameters of the truncated normal distribution for this case. These two solutions, the gamma and the truncated normal, which maximize the entropy under different constraints are tested as to their effectiveness for prediction of future events for homogeneous Poisson processes by measuring their coverage probabilities, the suitably normalized lengths of their prediction intervals and their goodness-of-fit measured by the Kullback–Leibler criterion and a discrepancy measure. The estimators obtained by these methods are compared in an extensive simulation study to each other as well as to the estimators obtained using the completely noninformative Jeffreys’ prior and the usual frequency methods. We also consider the problem of choosing between the two maximum entropy methods proposed here, that is, the gamma prior and the truncated normal prior, estimated both by matching of the first two moments and, by maximum likelihood, when faced with data and we advocate the use of the sample skewness and kurtosis. The methods are also illustrated on two examples: one concerning the occurrence of mammary tumors in laboratory animals taking part in a carcinogenicity experiment and the other, a warranty dataset from the automobile industry.
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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 | 0.006 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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".