Bayesian and Bühlmann credibility for phase-type distributions with a univariate risk parameter
Bibliographic record
Abstract
Credibility theory is a statistical tool to calculate the premium for the next period based on past claims experience and the manual rate. Each contract is characterized by a risk parameter. A phase-type (or PH) random variable, which is defined as the time until absorption in a continuous-time Markov chain, is fully characterized by two sets of parameters from that Markov chain: the initial probability vector and transition intensity matrix. In this article, we identify an interpretable univariate risk parameter from amongst the many candidate parameters, by means of uniformization. The resulting density form is then expressed as an infinite mixture of Erlang distributions. These results are used to obtain a tractable likelihood function by a recursive formula. Then the best estimator for the next premium, i.e. the Bayesian premium, as well as its approximation by the Bühlmann credibility premium are calculated. Finally, actuarial calculations for the Bühlmann and Bayesian premiums are investigated in the context of a gamma prior, and illustrated by simulated data in a series of examples.
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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 | 0.016 | 0.087 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.003 | 0.004 |
| 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".