Bronk Ramsey 2009 - Bayesian Analysis of Radiocarbon Dates - Overview
Summary
Bronk Ramsey (2009) shows that radiocarbon calibration is a single-event Bayesian update (likelihood from the calibration curve) and that multi-date chronologies are built by adding prior structure on event dates: order constraints, “uniform phase” group priors chosen so the prior on the phase span is neutral, and deposition models, all explored by Metropolis-Hastings MCMC in OxCal 4. The central warning is that the naive flat prior on every date strongly favours long spans (density ).
Overview
Research question. A 14C result is an isotope-ratio measurement, not a date; calibrated densities are multimodal and non-normal. How should many such results, plus stratigraphic and archaeological context, be combined coherently?
Contribution. A modular formulation: events on a continuous timescale; calendar information enters the likelihood, relative information (order, grouping) the prior; models are assembled from boundaries, phases and sequences; output and diagnostic tools (HPD ranges, agreement indices, simulation checks, outlier analysis).
Key results. (1) Standard calibration, Eq. 9 below. (2) Uniform-phase prior, Eq. 16, makes the span prior flat. (3) Agreement index with a rule-of-thumb threshold of 60%.
Motivating PyMC examples
E39 “How old is it? Radiocarbon calibration and Bayesian chronologies” (IntCal20 curve; Shroud of Turin measurements; Sluggan Bog core with Bchron) - example gallery. E39 finds that NUTS on a calendar date fails () because the posterior is multimodal, so the date is summed out on a 1-year grid; see Bronk Ramsey 2009 - Calibration Likelihood and the Calibration Curve.
Source Map
| Paper section | Pages | Note |
|---|---|---|
| Abstract, Introduction | 337-338 | this note, Bronk Ramsey 2009 - Calibration Likelihood and the Calibration Curve |
| Bayes’ theorem: the underlying methodology (events, calendar vs relative information, null prior, CQL) | 338-339 | Bronk Ramsey 2009 - Order Constraints and the Uniform Phase Prior |
| Calendar date information: timescale, PDFs, calibration Eqs. 3-11, Figures 1-2 | 339-342 | Bronk Ramsey 2009 - Calibration Likelihood and the Calibration Curve |
| Relative constraints, Eqs. 12-13, Fig. 3 | 342 | Bronk Ramsey 2009 - Order Constraints and the Uniform Phase Prior |
| Groupings: span problem, uniform phase Eqs. 14-16 | 342-345 | Bronk Ramsey 2009 - Order Constraints and the Uniform Phase Prior |
| Other distributions (ramped, exponential, normal) Eqs. 17-22, Figs. 4-5 | 345-347 | Bronk Ramsey 2009 - Other Group Priors Multiphase Models and Cross-Referencing |
| Multiple phases Eqs. 23-25, refinements Eqs. 26-30, Figs. 6-7, cross-referencing | 348-352 | Bronk Ramsey 2009 - Other Group Priors Multiphase Models and Cross-Referencing |
| Specific models: defined deposition rate (Eq. 32), other deposition models | 352 | Bronk Ramsey 2009 - Deposition Models and Sequences |
| Analysis methods (MCMC), model outputs, ranges, interrogation | 353-354 | Bronk Ramsey 2009 - MCMC Outputs Diagnostics and Agreement Indices |
| Diagnosis: testing models, simulations, Fig. 8, agreement indices Eqs. 33-39, MCMC problems, other problems | 354-358 | Bronk Ramsey 2009 - MCMC Outputs Diagnostics and Agreement Indices |
| Conclusions, references | 358-360 | this note |
Main Content
The five detail notes follow the paper. Calibration first (single event), then priors for several events (order, phase, others), then deposition-sequence models, then computation and diagnosis.
Connections
Survival Analysis (interval-censored events in time), Hierarchical Models (phase boundaries act like hyperparameters), MCMC Basics, Posterior Predictive Checking and Simulation-Based Calibration - Overview (the paper’s simulate-then-recover test is a forerunner of SBC).
Relevance to marketing measurement
The span-prior problem is the same trap as using flat priors on many exchangeable effects: the implied prior on a derived quantity (here the span, elsewhere total lift or max of effects) can be wildly informative. The paper’s check of running the model with no data to see what the prior alone does is a good prior-predictive habit (Prior Predictive Checking).