The Method of Forced Probabilities: a Computation Trick for Bayesian Model Evidence
Bibliographic record
Abstract
Bayesian model selection (BMS) can be used to objectively rank competing models of different structure and with different parameters upon comparison with validation data sets. This technique requires the evaluation of Bayesian Model Evidence (BME). BME is the likelihood of the data to occur under the assumed models, where the likelihood is averaged over the probability distribution of the model and its parameters. Exact and fast analytical solutions for BME exist only with strong assumptions. For that reason, other techniques and approximations for BMS/BME have been developed. While mathematical approximations via information criteria may suffer from strong biases in real-world applications, numerical methods do not rely on any assumptions but require high computational effort. This becomes prohibitive if the data set is very large, e.g. highly resolved in space and time. To still enable the use of BME as a probabilistic and rigorous model performance metric, we have developed the “Method of Forced Probabilities”: this method is a fast way to numerically compute BME for models that predict time series and fulfill the Markov Chain property in time. The core idea is to swap the direction of evaluation: instead of comparing thousands of forward runs of the model with the observed data (many model runs on random parameter realizations), we force the model to follow the data during each time step and record the individual probabilities of the model performing these exact transitions (single evaluation). As a test case for demonstration, we use invasion percolation (IP) models to simulate multiphase flow in porous media. The underlying, highly resolved data set was obtained from an experiment of a slow gas injection into water-saturated, homogeneous sand in a 25cmx25cm acrylic glass cell. Images were obtained at a rate of 30 images per second using the light transmission technique. Since IP models fulfill the Markov chain property, the Method of Forced Probabilities can be applied to evaluate their BME. Results confirm that the proposed method presents a scalable, inexpensive alternative to standard Monte Carlo methods for analyzing the model-data mismatch.
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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 | 0.009 | 0.050 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.004 | 0.003 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.002 | 0.008 |
| Insufficient payload (model declined to judge) | 0.014 | 0.004 |
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".