Bayesian calibration of a single date

The wiggles seen in the Suess curve were at first taken to be an artefact of mea-surement variability and calibrations were carried out by simple geometric means (eg. Ottaway and Ottaway 1972; Renfrew 1973). When, by the late 1970s, the radio-carbon community recognized that these wiggles are real and not just a statistical artefact, it became clear that calibration procedures needed to account for them. The simplest approach to the problem was to find the intercepts of the measurement and the calibration curve. Such an approach fails, however, when it comes to represent-ing uncertainties, as tryrepresent-ing to reflect them through the curve can cause paradoxes (Figure 3.13) (Dehling and van der Plicht 1993).

The Bayesian solution to this problem came in 1986 in the calibration program Calib (Stuiver and Reimer 1986). The procedure works because it inverts the question from what is the calibrated date range of this measurement? to what is the probability that a sample of this date would produce this measurement? such change in the direction

Figure 3.13: The paradoxes in radiocarbon calibration by direct geometric translation. Al-though the results on the left hand side are intuitively correct, geometric ap-proaches would result in the unrealistic results of the right hand side. In the upper row the case of calibrating against a change in the angle of the curve is pointed out and it is shown that under a geometric approach the calibrated date would have to be discontinuous. Below, the same is shown for a calibration plateau, where, according to geometry alone, the middle portion of the calibrated date range ought to be excluded. Re-drawn from Dehling and van der Plicht 1993, Figure 2-3.

of thought is the reason why until the mid-20^th Century Bayesian methodology was referred to as inverse probability (Fienberg 2006, 2). To be more specific, the algorithm begins with selecting a point on the calibrated axis. This axis is covered at that stage by a thin uniform probability distribution summing to unity. The algorithm then multiplies the value of this point by the corresponding products of the calibration curve and the measurement distribution and updates the calibrated date range to the value of this product. Hence, if the value of the measurement was approaching zero for a given point, the resulting value on the calibrated axis will also approach zero. If on the other hand the value on the measurement scale was high, so will be the value on the calibrated axis. Once all the points have been visited and the whole scaled to unity, the calibration is complete (Figure 3.14). In terms of Bayes theorem, the initial list of points on the calibrated time scale is the prior distribution, the measurement is the likelihood and the calibrated date range is the posterior distribution.

Figure 3.14: Bayesian calibration of a single radiocarbon date. 1) Begin on the low value of the vague prior distribution that covers the length of the calibration curve, 2) update your value by multiplying it through the likelihood function (here the curve and the age determination), 3) the updated posterior distribution is the calibrated date range. Hence, at point A, the posterior is negligible, as the value of the measurement was negligible, in point B it is small because the value of the measurement was small and in point C it is large because the value of the measurement was large.

This procedure affects the interpretation of the calibrated dates. First of all, standard deviations will not always describe the calibrated date well, and instead highest pos-terior density (HPD) areas are used (Figure 3.15). These areas are the regions under the posterior distribution which contain the greatest amount of probability within the smallest length of the abscissa (most often represented as the horizontal x-axis) (Hoff

2009, 42-3). HPD areas are often referred to in the radiocarbon literature as calibrated date ranges and throughout this thesis both terms will be used. The span and distri-bution of the HPD areas depends on the calibration curve. This dependency can be abstracted to three different situations (Weninger 1986):

1. The curve shape does not affect the shape of the calibrated date distribution.

2. A wiggle in the curve introduces bimodality in the calibrated date range. In terms of underlying processes this means that the amount of radiocarbon from samples of the two model ages is expected to be the same.

3. The amount of radiocarbon is expected to be similar for samples coming from an extended period, leading to the development of a plateau in the calibration curve stretches out the date range, leading to lower precision (Figure 3.16).

One such calibration plateau is the Halstatt calibration plateau which spans the pe-riod 750–400 cal BC and has substantial adverse effects on our ability to interpret archaeological sites from the earliest centuries of the Scottish Iron Age. For example, if two crannogs from that period are located next to one another in a loch, single ra-diocarbon dates will not be sufficient to determine whether they were built at the same time, within a few generations, or whether they were built several hundred years apart (Figure 3.17). It is the presence of this plateau and the large wiggle that follows that makes the radiocarbon-focussed part of the current project so important.

Figure 3.15: Highest posterior density areas. The shape of the calibration curve means that the calibrated date ranges are often asymmetric or multi-modal. Because of this the description of the results in terms of a mean and a standard deviation can be meaningless and hence highest posterior density areas are used instead.

Figure 3.16: A calibrated radiocarbon date from the Hallstatt plateau. Even though the two standard deviations of the age measurement cover only 120 years, the 95.4%

HPD region of the calibrated date spans more than 300 years.

Figure 3.17: Effects of the calibration curve on archaeological interpretation. In the sce-nario above two groups of simulated dates come from overlapping 50-year spans, whereas in the scenario below they come from 50-year spans set 275 years apart.

Nevertheless, the shape of the calibration curve during the Hallstatt plateau means that they cannot be distinguished.