Department of Probability and Statistics

Bayesian Archaeology

Since archaeological problems are typified by being relatively data poor and prior information rich, there are strong philosophical arguments for routine use of the Bayesian paradigm in archaeological research. Indeed, a number of researchers argued for adoption of such methods long before Bayesian statistics were being routinely used in other disciplines. As a result, archaeology was one of the very first applied areas to benefit from the recent developments in Markov chain Monte Carlo (MCMC) simulation techniques. Now that we can implement tailor-made models for a wide range of problem types, Bayesian methods are really coming into their own.

Bayesian methods are an important aid to archaeological data interpretation because we very often have relatively little data but considerable, informative prior information which is complex and hard to interpret heuristically. The Bayesian framework provides us with a formal set of tools for incorporating subjective a priori information into the interpretive process and, as a result, it has proved useful to specialists in a number of sub-disciplines of archaeology including the following.

• Estimating the radiocarbon calibration curve. Due to sun-spot activity and to a range of other less well understood events, the amount of radioactive carbon in our atmosphere has not remained constant over time. Thus, in order to convert radiocarbon determinations obtained from a radiocarbon laboratory into true calender dates, we need a calibration curve derived from radiocarbon determinations for known age samples. Such calibration data exist and have been collated and updated for more than 25 years. All members of Sheffield's Bayesian archaeology research group have recently worked on this problem and Caitlin Buck was the statistician on the IntCal04 team that put together the internationally-agreed radiocarbon calibration curve (for more on this see below).
• Interpreting radiocarbon data from archaeological and environmental research projects. Once the radiocarbon calibration curve has been estimated, there is still a great deal of statistical work to do in utilising the curve to help date groups of related archaeological and/or environmental samples. Members of our research group have undertaken collaborative work in this area for many years (see below for publications). We continue to develop and improve the statistical tools available by devising tailored models in response to particular problems encountered by the user communities. Currently, we have two PhD students working on such problems. Angela Howard has English Heritage and EPSRC funding for a project with the title Robust and Flexible Tools for Archaeological Chronology Construction (supervised by Caitlin Buck and Paul Blackwell). Lynsey McColl is funded by NERC/EPSRC under their Environmental Mathematics and Statistics initiative and is jointly supervised by Caitlin Buck, Paul Pettitt (University of Sheffield Archaeology Department) and Andrew Millard (University of Durham Archaeology Department). Her project has the title Statistical Tools for Investigating Issues of Contemporaneity in Palaeo-environmental and Archaeological Records.
• Field survey. Field survey is now a vital part of archaeological research. It includes use of techniques like geophysical surveying (ground resistivity, magnetometry, ground penetrating radar and the like), collection and analysis of soil samples or cores (for soil phosphate, pollen, chemical composition analysis and the like) and field walking (in which teams of archaeologists walk across landscapes recording surface finds such as pottery, architectural stone and flint tools). Bayesian methods have been shown to be of particular interest in the interpretation of particularly noisy field survey results, in particular soil phosphate data. Since a great deal of organic material (both animal and vegetable) contains phosphate, human activity (particularly sedentary agricultural activity) often gives rise to higher levels of phosphate in the soil than those which arise naturally. Unfortunately, however, most of the rapid techniques available for soil phosphate surveying give rise to data which are noisy, contain missing values and are often on quite a coarse scale. Typically, all archaeologists are wishing to do with such data is to assign cells in the survey grid as either associated with previous human activity or not. Even this level of interpretation, however, proves difficult using heuristic methods. Bayesian change-point methods, which allow for the inclusion of prior information about the likely levels of phosphate (both `on-site' and `off-site') in any given landscape, have been shown to have considerable interpretive power.
• Structural analysis. Much of prehistoric architecture in Europe is simple and has well understood structural properties (such as dry stone walling or mud bricks). In Greece and several other parts of Europe, however, there exists a class of structures with structural properties that are far less will understood. These are known collectively as corbelled domes, but the specific examples in Sardinia are called nuraghi and in Greece they tend to be called tholoi. These structures are, in fact, not strictly `domes' since the more modern technique of vaulting is not used to enclose the space. Corbelling must be undertaken with great care if it is to be stable as it involves the enclosing of a roof space by over-sailing courses of masonry until the space is spanned. For some time, architectural historians and archaeologists have been fascinated by these structures and have sought to understand how prehistoric peoples would have constructed them and made them so stable. There are a number of ways in which Bayesian statisticians could help in the investigation of such issues and one that has proved quite successful is to use change point analysis to identify possible locations for changes in the form or profile of a particular structure. It is now clear that not all prehistoric corbelled domes were constructed in the same way. More work is still needed before we will understand whether corbelled domes in different places have similar or different structures and the nature of any spatial structure involved.
• Chemical compositional analysis. Chemical composition analysis is now used quite widely by archaeologists to help them understand about things like pottery manufacture, soil composition and alteration, and to aid in the identification of forgeries. Sometimes such data can be interpreted quite easily without the need for statistical methods - especially in the case of very poor forgeries for example. In situations where we wish to group objects or soil types together on the basis of chemical composition, however, we can be dealing with large arrays of data and complex questions relating to the similarities between samples or groups of samples. Since the data are often noisy, are prone to missing values, and we sometimes have quite informative prior information about the nature of the groupings we would expect, Bayesian cluster analysis has been applied to data of this type with some success.
• Building relative chronologies. One of the best established uses of formal mathematical methods in archaeology is as a tool to aid in construction of relative chronologies on the basis of artefact types found during excavation (in particular of human burials). The techniques used to do this have become known as seriation and rely on the assumption that artefact types come into use, stay in fashion for a while and finally go out of fashion without ever appearing again in the archaeological record. Early formal tools for helping with identifying likely chronological orderings on the basis of this assumption were either deterministic or used non-tailored statistical tools. More recently, however, a relatively simple Bayesian modelling of the problem has been implemented using MCMC, allowing any prior information about orderings and structure to be included in the seriation process.

These are just a few of the archaeology projects that members of the Bayesian cluster have been involved with over the years. For more about on-going research in this area see below and for more on other archaeological and environmental work in the Department see the pages for our Statistical Modelling and Applied Statistics cluster.

Recent publications

An overview of developments in this area prior to 1996 can be found in

• Buck, C. E., Cavanagh, W. G. and Litton, C. D. (1996). The Bayesian Approach to Interpreting Archaeological Data, Wiley, Chichester.

Since then, members of our Bayesian cluster have been involved in a number of research projects in archaeology that can be summarised as follows.

Sample selection. In our sample selection work we sought to find ways to help archaeologists who had already undertaken a chronology building project and were now interested in dating new samples from those held in store. We were seeking to answer questions of the form: ``which samples should I send for dating next in order to optimize the quality of the chronology I build?''. The approach we chose was a risk analysis one which allowed us to trade gain in knowledge against the cost incurred in obtaining the new data. We implemented a set of simulation tools to allow us to estimate the level of knowledge that might be attained by sending different sets of samples for dating. The simulation code needed to do this is not that complex, but it does require large amounts of CPU time to run. The specific computing power needed is, of course, very dependent upon the precise problem under investigation, but the example we published in 1998 took enormous amounts of CPU time. We were able to shorten the real time taken to around 80 hours by using several workstations simultaneously, but this approach would not be available to very many archaeologists. One of the things we need to do in the future is to experiment with some of the more modern MCMC methods to see if we can speed things up.

• Christen, J. A. and Buck, C. E.(1998) Sample selection in radiocarbon dating, Applied Statistics, 47(4), 543-557.
  
• Buck, C. E. and Christen, J. A. (1998) A novel approach to selecting samples for radiocarbon dating, Journal of Archaeological Science, 25(4), 303-310.

Model choice. In our model choice work we identified some tools that might be useful to archaeologists interested in seriation (a method for relative chronology building described in our general description of Bayesian methods in archaeology). We developed some possible model-based approaches that might be used to aid in seriation and then implemented some MCMC methods (based on Langevin diffusion) to fit some of the proposed models. We then adopted predictive Bayesian model choice techniques to ascertain which of the models was most plausible for a given set of data and prior information. A summary of the statistical aspects of this work was published in 2000

• Buck, C. E. and Sahu, S. K. (2000) Bayesian models for relative, archaeological chronology building, Applied Statistics, 49(3), 423-440.

On-line MCMC. By the mid-1990s Bayesian methods for radiocarbon calibration had been appearing in both the statistics and archaeology literature for some time and we had developed quite a large number of practical software tools for helping archaeologists interpret their data. Given the CPU intensive nature of many of the methods, however, and the fact that few archaeologists have access to powerful computers, it soon became clear that we could not readily make our tools available to archaeologists simply by releasing code. With the increased popularity of the World-wide Web, we soon realised that one practical way to solve this problem was to write an on-line user interface to our code and to allow users from around the world to run their radiocarbon calibrations, via MCMC, on our server. With the help of a dedicated Java programmer for the duration of the project, in 1999 we launched the on-line Bayesian radiocarbon calibration service known as BCal. The software is described in more detail in our paper

• Buck, C. E., Christen, J. A. and James, G. N. (1999) BCal: an on-line Bayesian radiocarbon calibration tool, Internet Archaeology, 7,

Modelling the radiocarbon calibration curve. Due to the violation of the original assumptions of the radiocarbon dating method, a calibration process is required to transform radiocarbon-determined ages onto the calendar scale. A calibration curve linking calendar and radiocarbon ages is required in order to perform this transformation. Traditionally, the radiocarbon calibration curve has been modelled as the piece-wise linear curve joining the calibration data points, or by proposing cubic interpolation in order to obtain a smoother curve. Formally, the radiocarbon calibration curve can be seen as an unknown function that needs to be estimated from the calibration data, which are themselves subject to uncertainty. Within the Bayesian framework, we are able to make inference about the whole function, based on the information provided by the calibration data, and incorporating prior beliefs about important features of the curve, such as smoothness and differentiability. We model the radiocarbon calibration curve through a Gaussian process prior distribution on the space of all possible functions, specified according to our prior information regarding the underlying process of radiocarbon generation. The result is an estimate of the curve which adequately accounts both for the uncertainty associated with the curve itself, and for the uncertainty in the calibration data. Hence, in particular, our estimate of the calibration curve (the posterior mean) does not interpolate the data points. Moreover, the resulting variance values for the calibration curve seem more realistic than those resulting from other approaches.

• Buck, C. E. and Blackwell, P. G. (2004) Formal statistical models for estimating radiocarbon calibration curves. Radiocarbon, 46(3), 1093-1102.
• Gomez Portugal Aguilar, D., Litton, C. D. and O'Hagan, A. (2002) Novel statistical model for a piece-wise linear radiocarbon calibration curve, Radiocarbon, 44(1), 195-212.
  

Spatio-temporal modelling. Having developed a flexible, scalable and widely used framework for managing and interpreting temporal information from a single archaeological site, we are now keen to try to find ways to add a spatial component to our modelling structure so that we can begin to make a contribution to interpreting sites in relation to others in the same landscape or geographical region. There are many reasons why this is important, but perhaps the most obvious is that human activity in a given region or landscape is very rarely limited to individual sites or points. In practice it is not realistic to think of activity at each site as independent of activity at all other neighbouring sites in the same time period. Our conventional choice to interpret sites as if they were independent will have an effect on the temporal inferences we make and if these relate to regional or landscape events (such as the first arrival of a new community, technology or culture) it could have a marked impact on our ability to answer questions of real archaeological interest (like `which site in this region shows the earliest evidence of the new technology?', or `which region was colonised first A or B?'.) Members of our Bayesian methods in archaeology group are currently working on a number of research projects that we hope will help tackle questions like this. One paper has already been published in this area:

• P.G. Blackwell and C.E. Buck (2003), The Late Glacial human reoccupation of north-western Europe: new approaches in space-time modelling, Antiquity, 77, pp 232-240.

Tools for constructing chronologies. Building on the success of Bayesian modelling for the interpretation of radiocarbon data in archaeology. We are now broadening our horizons and thinking about the use of Bayesian methods for chronology building in general. Examples of recent output in this area include:

• C.E. Buck and A.R. Millard (eds), Tools for Constructing Chronologies: crossing disciplinary boundaries, Springer-Verlag, London
• C.E. Buck, T.F.G. Higham and D.J. Lowe (2003), Bayesian tools for tephrochronology, The Holocene, 13, 639-647.

PhD Topics

Research in this area would be supervised by Caitlin Buck, Paul Blackwell, Tony O'Hagan and/or one of the other members of the Bayesian Statistics Research Cluster. Please see individual staff pages for suggested topics.