Table 4.
Overview of Methods to Evaluate Bayesian Model Evidence
| Evaluation method | Abbreviation | Eq. | Underlying Assumptions | Comp. Effort | Performance in Linear Test Case | Performance in Non-linear Test Cases | Recommended Use |
|---|---|---|---|---|---|---|---|
| Analytical solution | |||||||
| Theoretical distribution of BME | - | 9 | Gaussian parameter prior and likelihood, linear model | Negligible | Exact | Not available | Whenever available |
| Normalizing constant of parameter posterior | - | 6 | conjugate prior, linear model | Negligible | Exact | Not available | Whenever available |
| Mathematical approximations | |||||||
| Kashyap's information criterion, evaluated at MLE | KIC@MLE | 14 | Gaussian parameter posterior, negligible influence of prior | Medium | Relatively accurate (assumptions mildly violated) | Inaccurate | KIC@MAP to be preferred |
| Kashyap's information criterion, evaluated at MAP | KIC@MAP | 15 | Gaussian parameter posterior | Medium | Exact (assumptions fulfilled) | Inaccurate | If assumptions fulfilled/ numerical techniques too expensive |
| Bayesian information criterion | BIC | 16 | Gaussian parameter posterior, negligible influence of prior | Low | Potentially very inaccurate (depending on actual data set), ignores prior | Not recommended for BMA | |
| Akaike information criterion | AIC | 18 | (not derived as approximation to BME) | Low | Potentially very inaccurate (depending on actual data set), ignores prior | Not recommended for BMA | |
| corrected Akaike information criterion | AlCc | 19 | (not derived as approximation to BME) | Low | Potentially very inaccurate (depending on actual data set), ignores prior | Not recommended for BMA | |
| Numerical evaluation techniques | |||||||
| Simple Monte Carlo integration | MC | 23 | None | Extreme | Slow convergence, but bias-free | Whenever computationally feasible | |
| MC integration with importance sampling | MC IS | 24 | None | High | Faster convergence, but (potentially) biased | As a more efficient alternative to MC | |
| MC integration with posterior sampling | MC PS | 25 | None | High | Even faster convergence, but even more biased (due to harmonic mean approach) | Not recommended for BMA | |
| Nested sampling | NS | 26 | None | High | Slow convergence for BME (due to uncertainty in prior mass shrinkage), but bias-free | Promising alternative to MC, more research needed | |