**About Bayesian Analysis:**

- Heralded as the “third radiocarbon revolution” (Bayliss 2009).
- Bayesian chronological modelling combines radiocarbon dates with information gleaned from the archaeological record (called
*priors*), such as stratigraphy, culture-historical frameworks, ceramic sequences, and settlement patterns (Bronk Ramsey 2009; Hamilton and Krus 2018; Lulewicz 2018). - Incorporating
*priors*with probability distributions from radiocarbon dates allow the analyst to use the relative ordering of samples in a stratigraphic sequence to constrain unmodelled calibrated date ranges to a narrower probabilistic range (modelled). - By making prior information explicit, Bayesian chronological models formalize assumptions and allow these to be tested through integration into different models.
- One of the main benefits of Bayesian analysis is an increase in the precision of dating of contexts when compared to calibrated radiocarbon dates that would otherwise only be considered individually.
- The analyst does not need an advanced understanding of statistics, but they must understand the concept of a model, and need to have a firm understanding of the contexts they are analyzing (Buck and Juarez 2015).

**Bayesian Software:**

- In general terms, Bayesian software packages (e.g., OxCal, CALIB, BCal) allow unordered groups of dates to be modelled within a
*phase*(an archaeological context), placed within a*sequence*(stratigraphy), and separated by*boundaries*(start/end ages for specific archaeological contexts). - For our study, we used OxCal 4.4, which is freely available on the internet. This software has a simple, easy-to-use, interface that separates the analyst from the complex mathematical details that underpin Bayesian analysis.
- OxCal uses Markov Chain Monte Carlo (MCMC) algorithms to approximate all possible solutions and probability outcomes.
- Agreement indices are generated for posterior distributions of each radiocarbon date in a model (
*A*) and for the model in its entirety (*A-model*). - Outlier dates and/or problematic assumptions (priors) are identified by agreement indices that fall below a critical value (
*A’c = 60*), which indicates the data has a poor fit in the model (analogous to 0.05 significance level in a χ^{2}test). - Importantly, even if the agreement index is above
*A’c*this does not necessarily mean that the model is “correct”; instead, this value simply indicates that there is good fit between the model and the data. - As with all statistical analyses, the outcomes of Bayesian models are only as reliable as the assumptions that go into them.
- The following OxCal commands were used in this study (table adapted from Culleton et al. 2012:1577, Table 3)

OxCal Command | Stratigraphic Context |

Phase (Unordered Group) | Multiple dates within a fill Multiple dates within a single habitation debris context or a single occupation surface Groups of dates separated by a common stratigraphic marker |

Sequence (Order Group) | Dates separated by series of clearly defined stratigraphic contexts (e.g., floors) Dates on materials within well stratified middens |

Boundaries | Events that bracket the beginning and end of a phase but are not directly dated (e.g., clearing or levelling a site before construction; end of construction; abandonment of structure) |

**Regular Models (Uniform Models) vs. Charcoal Outlier Models:**

- Default Bayesian models in OxCal use a uniform prior distribution (UPD).
- This assumes that any event in the model is equally likely to have occurred in any individual year covered by the data (Bronk Ramsey 1998:470).
- Uniform Prior Distributions structure data as a continuous period of activity.

- Outlier models, in contrast, tell the system to assume a normal distribution instead of the standard, uniform model (Bronk-Ramsey 2009).
- The underlying premise of a Charcoal outlier model is that “old” wood dates can be better integrated into the sequence by using an exponential curve, wherein “old” dates create a curving influence on the others, pushing them towards the ends of their distribution while remaining within the parameters of the model (Hanna et al. 2016:783).
- This is coded in OxCal as (
*OutlierModel (N(0,2),0,“t”)*:*t*indicates the timing the event could be wrong and*N*indicates a normal distribution with a mean of 0 and a standard deviation of 2 (Hanna et al. 2019). This code provides probability on whether a given date is an outlier, provided all samples are assigned the designation*Outlier (0.05)*. When designed as a possible*Outlier*, the system downweights those measurements that disagree most with the others. 0.05 is used because the model assumes a 1 in 20 chance that charcoal measurements require shifting (Bronk-Ramsey 2009:1026).

**Alabama Data**:

Outlier Model Results

Regular Model Results

OxCal Code

**References:**

Bayliss, Alex (2015). Quality in Bayesian Chronological Models in Archaeology. *World Archaeology* 47(4):677–700. DOI:10.1080/00438243.2015.1067640.

Bronk Ramsey, Christopher (1998). Probability and Dating. *Radiocarbon* 40(1):461–474.

Bronk Ramsey, Christopher (2009). Bayesian Analysis of Radiocarbon Dates. *Radiocarbon* 51(1):337–360. DOI:10.1017/S0033822200033865.

Buck, Caitlin E., and Miguel Juarez (2017). Bayesian Radiocarbon Modelling for Beginners. *arXiv:1704.07141 [stat]*. accessed November 23, 2021.

Culleton, Brendan J., Keith M. Prufer, and Douglas J. Kennett (2012). A Bayesian AMS 14C Chronology of the Classic Maya Center of Uxbenká, Belize. *Journal of Archaeological Science* 39(5):1572–1586. DOI:10.1016/j.jas.2011.12.015.

Hamilton, W. Derek, and Anthony M. Krus (2018). The Myths and Realities of Bayesian Chronological Modeling Revealed. *American Antiquity* 83(2):187–203. DOI:10.1017/aaq.2017.57.

Hanna, Jonathan A., Elizabeth Graham, David M. Pendergast, Julie A. Hoggarth, David L. Lentz, and Douglas J. Kennett (2016). A New Radiocarbon Sequence from Lamanai, Belize: Two Bayesian Models from one of Mesoamerica’s Most Enduring Sites. *Radiocarbon* 58(4):771–794. DOI:10.1017/RDC.2016.44.

Lulewicz, Jacob (2018). Radiocarbon Data, Bayesian Modeling, and Alternative Historical Frameworks: A Case Study From the US Southeast. *Advances in Archaeological Practice* 6(1):58–71. DOI:10.1017/aap.2017.29.