3. Bayesian Chronological Modelling
Alex Bayliss and Peter Marshall
Bayesian statistics provide an explicit, probabilistic method for combining different sorts of evidence to estimate formally the dates of events that happened in the past. The basic idea is encapsulated in Bayes’ theorem, which simply states that we analyse the new data we have collected about a problem (“the standardised likelihoods”) in the context of our existing experience and knowledge about that problem (our “prior beliefs”). This enables us to arrive at a new understanding that incorporates both our existing knowledge and our new data (our “posterior beliefs”). This is not the end of the matter, however, since models will be updated as new information becomes available.
At its most basic, this approach simply takes account of the fact that a group of dates are related in some way, for example by being from the same site or associated with the same type of artefact. It is essential to account for this in the analysis of any scientific dates, or there is a significant risk that past activity will be interpreted as starting earlier, ending later, and enduring for longer than was actually the case (Bayliss et al. 2007). This is because the probabilistic date estimates provided by a range of scientific techniques ‘scatter’ around the actual age of the sample; and this scatter matters (Bayliss and Marshall 2022, section 2.1).
Figure 10 illustrates this using the assemblage of radiocarbon dates on ultra-filtered gelatin extracted from human and cut-marked animal bones found in Gough’s Cave, Somerset (Table 1; Jacobi and Higham 2009; note that following their interpretation, OxA-18067 has been excluded as this related to later activity).
TABLE 1: Radiocarbon ages and associated measurements on ultra-filtered gelatin from Gough’s Cave, Somerset (see Jacobi and Higham 2009, table 1 for further measurements from this site)
| Laboratory Code | Material and context | Radiocarbon Age (BP) | δ13CIRMS (‰) | δ15NIRMS (‰) | %C | C/Natomic ratio | Gelatin yield (mg) |
|---|---|---|---|---|---|---|---|
| OxA-18065 | M.49797, Equus ferus, cut left 1 phalange from Layer 8 of R F Parry’s excavation (1927–31) | 12,490±55 | −20.5±0.2 | 1.6±0.3 | 43.2 | 3.2 | 26.2 |
| OxA-17845 | M.49758, Cervus elaphus, cut 2nd phalange from Layer 11 of R F Parry’s excavation (1927–31) | 12,500±50 | −19.6±0.2 | 2.8±0.3 | 47.4 | 3.2 | 37.3 |
| OxA-17848 | 1.1/4, adult human calotte conjoined to frontal (GC 1987 169) from Layer 12/13 of R F Parry’s excavation (1927–31) | 12,485±50 | −19.3±0.2 | 8.5±0.3 | 49.7 | 3.2 | 11.8 |
| OxA-16378 | M.49847, Cervus elaphus, cut distal right metatarsal from Layer 13 of R F Parry’s excavation (1927–31) | 12,515±50 | −19.8±0.2 | 3.2±0.3 | 43.7 | 3.2 | 28.8 |
| OxA-13585 | M.49877, Canis cf familiaris, right dentary from Layer 14 of R F Parry’s excavation (1927–31) | 12,440±55 | −18.5±0.2 | 5.8±0.3 | 54 | 3.5 | 26.3 |
| OxA-17833 | M.49955, Equus ferus, cut right 2nd phalange from Layer 14 of R F Parry’s excavation (1927–31) | 12,570±45 | −20.7±0.2 | 1.1±0.3 | 43.7 | 3.2 | 53.5 |
| OxA-17832 | M.50024, Equus ferus, cut distal right metacarpal from Layer 18 of R F Parry’s excavation (1927–31) | 12,415±50 | −20.9±0.2 | 1.5±0.3 | 43.8 | 3.2 | 42.4 |
| OxA-12104 | M.50048, Equus ferus, righ M1/M2 from Layer 24 of R F Parry’s excavation (1927–31) | 12,495±50 | −20.6±0.2 | 1.0±0.3 | 42.5 | 3.1 | 30.6 |
| OxA-17847 | M23.1/2, human, cut right scapula from lip of ‘Cheddar Man Fissure’ (1959) | 12,565±50 | −19.0±0.2 | 7.9±0.3 | 45.2 | 3.2 | 42.1 |
| OxA-18067 | GC 1986 1, Cervus elaphus, cut distal right tibia from top of temporary section on western edge of ‘Cheddar Man Fissure’ (1986) | 12,245±55 | −20.2±0.2 | 2.6±0.3 | 42.8 | 3.2 | 51 |
| OxA-18066 | GC 1986, 27A, Lynx lynx, cut shaft of left femur from base of temporary section on western edge of ‘Cheddar Man Fissure’ (1986). | 12,440±55 | −19.3±0.2 | 4.8±0.3 | 43.2 | 3.2 | 15.8 |
| OxA-17849 | GC 1987 190, adult human cut calotte from Area I of the Natural History Museum excavation (1987–9) | 12,590±50 | −19.3±0.2 | 7.7±0.3 | 50.4 | 3.1 | 51.4 |
| OxA-17846 | GC 1987 25, bevel-based rod of Mammuthus primigenius ivory from Area I of the Natural History Museum excavation (1987–9) | 12,470±55 | −21.2±0.2 | 6.8±0.3 | 48.4 | 3.2 | 9.4 |
| OxA-18064 | GC 1989 99, bâton percé of Rangifer tarandus antler from Area I of the Natural History Museum excavation (1987–9) | 12,535±55 | −19.2±0.2 | 1.8±0.3 | 42.5 | 3.2 | 56.2 |
| OxA-18068 | GC 1987 191, Equus ferus cut cervical vertebra from Area I of the Natural History Museum excavation (1987–9) | 12,520±55 | −20.1±0.2 | 3.1±0.3 | 42.8 | 3.2 | 52.8 |
| OxA-16292 | GC 1987 187, Equus ferus cut cervical vertebra from Area I of the Natural History Museum excavation (1987–9) | 12,585±55 | −19.8±0.2 | 0.4±0.3 | 41.9 | 3.2 | 19.8 |
In this graph the ‘raw’ scientific dates are shown in outline, and the posterior beliefs from the Bayesian model are shown in black. Some posterior distributions relate to particular objects. For example, cut-marked bone GC 1990 184 dates to 15,010–14,820 cal BP (93% probability; OxA-18035; Figure 10) or 14,680–14,630 cal BP (2% probability), probably to 14,950–14,860 cal BP (68% probability).
Other posterior distributions estimate the time of events in the past that do not relate to a particular sample. For example, this model estimates that human occupation in the cave began in 15,060–14,850 cal BP (93% probability; StartGough’sCave; Figure 10) or 14,680–14,650 cal BP (2% probability), probably in 14,980–14,890 cal BP (68% probability).
Date ranges deriving from Bayesian modelling are conventionally given in italics to distinguish them from unmodelled scientific dates. They should be cited with the relevant parameter name and a reference to the model from which they derive.
Archaeologists have a whole range of other information that can be included as prior information in Bayesian models.
Relative dating can be provided by typological analysis of artefacts or, most commonly, by stratigraphy. This stratigraphy can be within a single site (see Gransmoor) or within the geomorphology of sets of related features, such as river terraces (see The Axe Valley at Broom).
Often an archaeological site, considered in isolation, will provide limited new evidence on a particular issue. However, this evidence can contribute to larger questions in an updated chronological model of the problem at hand (see Marine Aggregate Licence Area 240).
The need for constant revision and rebuilding of Bayesian chronological models means that a report on chronological modelling must not only explain and justify the models presented, but also provide sufficient information to allow them to be criticised and reconstructed in the future.
Reports should include:
- Objectives of the study: including the dating precision needed to achieve the objectives and how the objectives may have been (re)cast in the light of the available material, prior information, funding, etc.
- Methodology: including a statement of the approach adopted and the statistical methods and software used.
- Sampling strategy: including a discussion of the selection of the scientific dating techniques employed; the available prior information; the available pool of potential samples; the results of any simulation models; and the rationale by which these elements have been combined into a strategy.
- Details of scientific dates: see the appropriate sections of these guidelines for the information required for different techniques.
- Model definition and description: each model must be explicitly defined so that it can be reproduced. Most models can be defined using procedures provided by publicly-available software packages, although models that use new statistical procedures will need mathematical appendices. Prior information should be described, and its strengths and weaknesses assessed; the robustness of the associations between the scientific dates and the prior information should be considered; the compatibility of the scientific dates with each other and with the prior information should be assessed; outliers or misfits should be identified and described.
- Sensitivity analyses: alternative models, which vary components of a model to determine how sensitive the modelled chronology is to changes in the interpretations on which it is based.
Further information on Bayesian chronological modelling can be found in Bayliss and Marshall (2022).
