calibration of radiocarbon results pertaining to related archaeological events

Calibration of Radiocarbon Results Pertaining to Related Archaeological Events

C. E. Buck’, C. D. Litton” and A. F. M. Smithh

(Received 10 May 1991, revisedmanuscript accepted 3 September 1991)

With the advent of the high precision radiocarbon calibration curve there is an increas- ing demand from archaeologists for results, previously reported on the radiocarbon scale, to be expressed on the calendar scale. Furthermore, there is a general realization within the archaeological community that much more information can be derived from radiocarbon dating if the samples are taken from coherent, well-chosen contexts and if any available substantive archaeological information is included in the data analysis. This paper describes how the Bayesian approach to statistical data analysis can be used for the calibration of groups of radiocarbon results in situations where archaeological information is available a priori about the relationships between the contexts being dated. Recent innovations in the implementation ofthe Bayesian paradigm are used and the methodology is illustrated by the analysis of data arising from two archaeological excavations.

Keywords: RADIOCARBON CALIBRATION, BAYESIAN STATISTICS. GIBBS SAMPLER.

Introduction The high precision radiocarbon calibration curves of Pearson & Stuiver (1986) and Stuiver & Pearson (1986) are now widely known and accepted and there is an increased demand from archaeologists to be able to quote, and make use of, dates on the calendar scale. The CALIB program (Stuiver & Reimer, 1986), now generally available, makes use of the high precision curves and provides a straightforward means by which archaeologists can calibrate individual radiocarbon results or groups of results relating to the same calendar date. As far as we are aware, no readily available computer program has been produced that allows archaeologists to calibrate radiocarbon results corresponding to two or more related calendar data simultaneously. In a recent paper, Aitchison et a/. (199 1) describe a method for the estimation of a,floruit of an archaeological site on the basis of a number of radiocarbon results, but their method does not allow for the inclusion of any archaeological information about the relationships between the events under consideration. If an approach that allows the combining of radiocarbon results with archaeological information can be found, then we will have increased confidence in the results and subsequent interpretation. No longer will we be restricted to calculating individual dates

“Department of Mathematics, University of Nottingham, Nottingham NG7 2R 0. U.K. “Department of Mathematics, Imperial College. London SW7 2BX. U.K.

497

0305- 4403’92 0504971 I6 $08.00:0 0 1992 Academic Press Limited

49x c’. E. HU<‘K ET Al..

and then asking whether the results accord with the archaeological evidence. Instead the archaeological evidence will be included in the analysis.

The wide availability of CALIB has coincided with improvements in the precision and accuracy of the radiocarbon results quoted by the laboratories. In addition to this, radiocarbon laboratories have been taking an increased interest in advising archaeologists about the most effective sampling strategies to adopt (Bowman, 1990: 62). This arises from the realization that much more information can be derived from coherent, well-chosen groups of radiocarbon results and associated archaeological information than from individual (or even multiple) results for one event (or context) alone. Adoption of a coherent sampling strategy is the only way that archaeologists can make best use of radiocarbon dating and still keep within the tight financial constraints of most modern excavation budgets.

In this paper we consider the calibration of groups of radiocarbon results in situations where there is u priori information available (or postulated) about the archaeological relationships between the contexts being dated. We consider three different cases in which we wish to be able to make inferences.

(i) The calibration of radiocarbon results when good quality archaeological information is available about the likely chronological ordering (or partial ordering) of the contexts being dated. Given the radiocarbon results and the information about ordering. we require inferences about the calendar dates of the contexts and, possibly, the length of time elapsed between the various events represented by the contexts.

(ii) The calibration of radiocarbon results when good quality archaeological information is available about the relationship between phases represented on the site. In this situation we require a means for inferring the beginning and ending dates for the phases.

(iii) In the situation where the archaeologist is unsure about the chronological relationship between the various contexts for which there are radiocarbon results, we seek a means for testing the relationships between the dates as represented by the radiocarbon results. For example, we might require information about the relationship between several phases on the same site; are the phases abutting. overlapping or widely spaced in time’?

This is clearly not a complete list of all situations in which we might wish to combine LI prioriarchaeological information with radiocarbon results. Readers with a background in archaeology will readily think of other applications and in doing so we hope that they might be encouraged to adopt the approach advocated here, which allows the combining of both types of information in the same statistical analysis.

Statistical Modelling We begin by considering the relationship between a single radiocarbon result and the calendar date of the corresponding archaeological event. We assume that the event under consideration occurred at calendar date 0, measured in years BP. which is unknown. Associated with this unknown calendar date is a unique “radiocarbon date”, which is related to the amount of ‘%Y actually present in the sample. Since the radiocarbon date is dependent on the calendar date we represent it by p(O). Due to the nature of the samples available for radiocarbon dating and the chemical and physical techniques used to arrive at the radiocarbon result, the r.~~)erimetlr~~/l~~ derived values available for ~(0) are not totally accurate or precise. What we are provided with is an observation, J‘ (a specific numeric value), which is an estimate of p(0). The actual value ~3 is one of many possible Lalues that could have been arrived at; each actual measurement will produce a different result because of statistical variability. In statistical terms J’ is a realization of a random variable, the latter conventionally denoted by 1’. which can be expressed ah

CALIBRATION OF RADIOCARBON RESULTS 499

Y = p(O) + noise,

where the statistical noise term represents these inherent inaccuracies and imprccisions in the measurement process. The noise, or error term, is conventionally assumed to have a normal (Gaussian) distribution, with a mean ofzero and a variance represented by IS’. The distribution of the random variable is, therefore, normal with mean ~(0) and variance 02, kvhich is written as

Y - N(p(Q),02).

The next task is to relate ~(6) to 0. This is achieved using the high precision calibration curve expressed in its piecewise linear form

a, + 6,8 (0 d t,,) cl(~) =

i uk + h,O (tL-, < H 6 t,,k = I,2 ,..., K) UK + h&l (0 1 fa),

where the /, are the knots of the calibration curve, K+ I is the number of knots used and where LI, and hl are assumed to be known constants. Then we have

y - N(u, + 0,8,0’) (0 < f(j) Y - N(u, + h,O,o’) (fl, / < 8 6 t,,k = I,2 ,..., K) Y - N(LIK + h,El,o’) (tk. < 0).

We now take into account that, in practice, several samples are often taken for radiocarbon dating from the same context and so should be of the same calendar date. Suppose we have a set of I radiocarbon results related to the sume event of calendar date 8. Let these radiocarbon results be represented by random variables Y,. Y,,.... Y,. These must all have the same expectation. p(O), but possibly different variances denoted by o~.&....o$ Statistical theory establishes that these I results may be combined into a weighted mean (see Ward & Wilson. 1978) by letting

which can be shown to have , /

! 1

-I expectation ~(0) and variance ~~ = 1 CT, 2

1-l

Finally, WC extend our model to the situation where we have radiocarbon results X,,X,,....X,, (each possibly composed of more than one result combined into an overall summary as above) related to t7 individual events, with event i having unknown calendar date 0,. Assuming that. conditional on the calendar dates 0,,f3,,...,8,2. the radiocarbon results are independent, then the joint probability density function of X=(X,...~li,,) is given by

Inference About the Calendar Dates In this section we describe the framework for Bayesian statistical infercncc as applied to our problem. More general discussions of the applicability of the Bayesian paradigm to

so0 Cl’. E. BUCK ETAL.

archaeology are given in Orton (1980: 220) and Ruggles (1986). For discussion of the applicability of Bayesian methods to radiocarbon calibration in particular see Litton & Leese (1991) and Buck et ul. (1991).

Here we are interested in making inferences regarding n calender dates Q,,02,...,0,Z. Given these dates we have a statistical function (called the likelihood function), p(x)El,z), which relates the data .x,,.Y~ ,...,. Y,, to the unknown calendar dates 8,,Bz ,..., 8,. In any real situation, we also have some prior belief about the 8s which can be expressed in terms of a prior density, p(O) where f~=(Q,,...,0,). We discuss below how this prior density is determined. The Bayesian paradigm involves calculating the posterior density of 8, p(Qx), which combines the archaeological prior information and the information provided by the radiocarbon results. The relationship between the posterior density, the likelihood and the prior density is given by Bayes’ theorem in the form

This formula summarizes the way in which our prior beliefs are modified by taking into account the data we have observed. (In practice, the denominator can effectively be ignored as it is just a normalizing constant.)

To make inferences regarding the calendar dates we must now elicit from the archaeologist the relevant information regarding the contexts of the samples and how they are believed to be related chronologically. In turn this information will be expressed mathematically in terms of the prior density for 8, denoted byp(B). For example if we have one radiocarbon result corresponding to calendar date 8 and nothing is known about this date, we would take p(O) to be equal to a constant. On the other hand, if archaeological evidence suggests a terminus unte quem, El,, and a terminus post quem, Q,,, for the calendar date, then the prior for 8 will be taken as

pm = i

(e, - e,)- 1 8, > 8 > ep 0 otherwise.

Consider the situation in which there is clear archaeological evidence, perhaps based on the stratigraphy of the site, that the events to be dated are ordered that is, event I precedes event 2 which precedes event 3 etc. Apart from this, no other information exists regarding the calendar dates 8,....,8,. In this case, the prior density for 0 could be taken as

m) = c 8, > 8, > > e,, 0 otherwise,

where (’ is a constant. Whatever the situation, from the joint posterior density of 0, the marginal posterior

densities of the 8, need to be calculated. In the first example given above, of one result corresponding to a calendar date 8 with no prior information about 0, the posterior density of the calendar date 0 conditional on the radiocarbon result is

and so

In other words. within the interval ({,- ,,tA), the posterior distribution of 8 is normal with mean D;‘(Y-~1~) and variance h,?r-. This is, in fact, the mathematical representation of the familiar form produced by CALIB.

<‘ALIBRATION OF RADIOCARBON RESULTS 501

However, in general, technical difficulties that arise in the calculation of the marginal densities, particularly for large values of n, have impeded the wider application of Bayesian techniques to data analysis. Apart from a few, relatively simple, special cases, the calculation of the marginal densities and corresponding sample statistics (e.g. mean, standard deviations) require the use ofadvanced techniques ofnumerical analysis in order to carry out the necessary multidimensional integration involved. Over the last few years there have been a number of advances in numerical approximation techniques for such calculations; for example, Naylor & Smith (1982), and Smith et ul. (1985, 1987). These techniques are typically for use by the statistical specialist and/or require sophisticated software. However, in a recent paper, Gelfand & Smith (1990) describe the use of the Gibbs sampler to evaluate posterior densities. This is, we believe, a major breakthrough in the use of numerical techniques for Bayesian data analysis as it provides a simple framework within which to calculate posterior densities. Examples of its use in a variety of situations are given in Carlin et rtl. (1991) and Gelfand et ul. (1990).

The Gibbs Sampler The Gibbs sampler is an iterative simulation technique, which effectively side-steps the need for sophisticated numerical expertise in implementing Bayesian methods. If (P,,(P?,...,(P~ are unknown parameters about which we wish to make inferences via their posterior distributions, the algorithm is based upon sampling, using simulation in a systematic fashion, from the conditional densities as follows.

(i) Choose some arbitrary starting values (p(~),q~$~....,(p$‘~.

(ii) Sample cp(l) f rom the conditional density of cp, given (p$“,...,(p(~) and the data. Sample $1) from the conditional density of ‘p? given cp$‘),(p(:),...,(p$? and the data.

Sample cp(i) from the conditional density of (pk given cp$‘),cp(~‘,...,cph’~, and the data. This completes one iteration of the algorithm.

(iii) Repeat (ii) Y times with (p(y)and cp(,‘)replaced by cp(:-‘i and cp’,‘) respectively t = 2,3,...,r.

Under fairly general conditions, Geman & Geman ( 1984) show that the distribution of the sampled values cp(:) tends to the distribution of ‘p,. This algorithm is repeated m times, possibly using different initial values. Following Gelfand & Smith (1990) it is extremely easy to estimate the marginal density of ‘p, as follows. Firstly we note that the rth iteration of thejth replication we have observed values (pl’,),H’z;‘.....,Otl. Conditional on cpjy)(l# i) we can write down the density of cp,. The marginal density o i- cp, IS estimated by averaging the conditional density of cp, conditional on @(l#i) over replicates 1,2,...,m. In the subsequent illustrative examples this is carried out over a discrete grid and, as a consequence, a “histogram-like” summary of the marginal probability distribution is produced (suitable for interpretation by archaeologists).

Illustrative Examples We will now consider the three cases outlined in the introduction using examples from two archaeological excavations. We consider case (i) using radiocarbon results and archaeological information from excavations at Runnymede Bridge under the direction cf Dr S. Needham. Examples of cases (ii) and (iii) are illustrated using radiocarbon results and archaeological information from excavations at Danebury under the direction of Prof. B. CunIiffe.

502 C. E. BUCK ET/IL.

Event Calendar

date

Number Combined of radiocarbon

samples result

Clcarancc f), Waterfront I 0, Waterfront 2 0; Hardstanding 0,

5 2729+32w 4 2742 +2X BP 4 2655 i 30 HP 2 2519*35 RP

Runnymede Bridge Full details of the archaeology and radiocarbon dating of the Neolithic and Bronze Age phases at Runnymede Bridge are given in Needham (199 I ) and are not included here. We offer only an outline of the information immediately relevant to our example. We are concerned here with the radiocarbon results of samples taken from the Late Bronze Age phase in the area known as Area 6. The samples for dating were taken in an attempt to provide calendar dates for four distinct archaeological events. These events were (1) a period of clearance of vegetation, (2) the driving of piles into the river bed to form a water front, (3) the driving of further piles to form a second water front and (4) the construction ofhard standings outside the outer pile row. The samples were mainly ofwood, although a few were charcoal, and great care was taken to ensure that they were contemporary with the event to be dated. The combined radiocarbon results for each event are given in Table 1.

Denoting the calendar dates BP of the four events by 0,. O,, 8, and 0, respectively we can then represent any archaeological information available to us about the relationship between the calendar dates in terms of 0s. At Runnymede Bridge there is archaeological evidence which firmly suggests that the period of clearance precedes all other events. The clearance period is short and so is seen as one event. We have no firm archaeological evidence about the relationship between the calendar dates for events 2, 3 and 4; apart from the assumption that they are later than event 1. We can also make the further assumptions that the oldest possible calendar date for any of the four events is O,, and the youngest 0,. so that

8, > 8, > 8, > 8,; 8, > o1 > 0,; 0, > 0, > 8,.

Thus the prior information regarding 8,. c)?, 0, and 8, is modelled as

1’(0,,~&8J c = 0” > 0, > 0, - > OS; 0, > 8, > 0s; 8, > cl4 > 05 0 otherwise.

where C’ is constant. Straightforward inspection of the mathematical forms involved establishes that the conditional density of 8, given 02, 8, and 0, is normal with mean hi ‘(.u, -(I~) and variance h;‘? for t, , CO< t,, with the additional restriction that max(f3,,0,,0,)<8, ~0,~. The conditional density of O,(i=2,3,4) give 0, and O,(j,~i) is normal with mean h, ‘(X,-Q,) and variance h;‘? for t, , <8< t,, with the addttronal restriction that 8,<0,<8,. Given this formulation, it is relatively straightforward to sample in turn from the conditional distributions of Cl,, O,, 8, and Q4.

Using this model (with 0,,=3005 and 8, =2305-chosen to give a broad range of possible dates for O,,...,O,) and the Gibbs sampler methodology outlined above with


8, 0.2 -

0.0 I - I I I 3050 2950 2850 2750 2650 2550 2450

0.4 4

2 0.2 - - E E 0.0 e 3050 2950 2850 2750 2650 2550 2450 a k5 0.4

.i a, 4 t 0.2- B

0.0; I 1 I I 3050 2950 2850 2750 2650 2550 2450

0.4 04

0.2 -

o-0 I I I_ 3050 2950 2850 2750 2650 2550 2450

Callbroted date BP

Figure I, Posterior probability plots for the calibrated dates BP of~he Ihur ecents at Runnqmcde Bridge.

177 = 200 replications and I’ = 50 iterations, we obtain “histogram-like” summaries of the posterior distributions of the calendar dates of the individual events and of the length of time elapsed between events. Figure 1 shows posterior probability plots for the calibrated dates BP of the events given our model and Figure 2 shows estimates of the associated posterior probabilities of the length of time between the events. Table 2 shows the 95% highest posterior probability regions for the four events based upon the results in Figure I. These intervals are formed by selecting the lo-year periods with the highest posterior probabilities until the sum of the probabilities of the periods included in the intervals reaches 0.95. In this manner. any IO-year period included in a reported highest posterior probability region has a higher posterior probability than any IO-year period outside the region. Another consequence is that the intervals so formed are the shortest that can be found for our fixed percentage of total probability and thus provide the “sharpest” possible inferences.

As can be seen from Table 2. the 95% highest probability region for event 1 contains the periods 2930 to 29 10 and 2890 to 2790 cal BP. (The fact that there are two “likely” periods and not just one is a consequence of the “wiggles” in the calibration curve, which result in several calendar periods being associated with the one radiocarbon result.) The reader familiar with betting language can interpret these intervals by saying that the posterior odds are 19 to 1 against the true value of 8, not lying in one of these two intervals.

Considering Figure 2, we can see that, although we have assumed that 8, >O,, UC: nonetheless arrive at posterior estimates for the length of time between the clearance and waterfront I which suggest strongly that the dates for the two events are indistinguishable. The 95% highest posterior probability range for the length of time between event I and

SO4 C. E. BUCK ET AL.

)O

0.2

0.15 -

0.1 -

0.05 - I

200 4c 2 of time between and .- Length 8, 8, (calendar years) .z 0.2. 13

z 0.15 -

h -r--

0.1 -

Length of time between e2 and@, (calendar years)

-400 -200 0 200 400

Length of time between 6’S and 8, (calendar years)

Figure 2. Posterior probabilities of the length of time between the events at Runnymede Bridge.

Event 95% highest posterior probability region

I 2930 to 29 IO cal BP 2880 to 2790 cal BP

2870 to 2780 cal BP

2840 to 2820 cal BP 2X 10 to 2750 cal BP

1780 to 27 IO cal BP 2630 to 2610 cal BP 2580 to 2540 cal BP 2530 to 25 10 Cal HP

event 2 is 0 to + 70 years with the mode of the distribution towards the early part of the region. The similar region for the length of time between event 2 and event 3 is 0 to + 100 years and for the length of time between event 3 and event 4 is - 10 to + IO0 years, + 140 to + 160 years and + 190 to + 250 years. Thus it is highly likely that event 2 precedes event 3 which in turn precedes event 4.

Here we have modelled “events” as occurring at precise dates. In some circumstances, however, the archaeologist might believe that the “events” under consideration occurred over a period of time. In this situation we would require a more sophisticated model of the type described in the next section.

Radiocarbon result BP

Ceramic phase

Harwell number

Radiocarbon result tw

Ceramic phase

2029 2032 2033 3039 3085 2567 2581 2585 3026 3726 4464 2030 4325 4327 432X 4329 4330 4331 4327 3470

964 965

1442 7037 2038 2971 3021 3022 4243 4244 4278 4279 4339 4343 4466 1368

2530+110 2370+X0 2460*60 242Oi80 2440*70 2210+60 2160+80 2330&60 245Oi80 227Oi70 2300+70 229OF60 2270+90 2120+70 233Ok90 214Ok80 258Oi80 2340+100 2300:90 218Ok90 223Ok70 2210+70 209Oi90 2060 * 80 209Oi70 2110*70 221Ok70 2210&70 2250*70 2120+80 2470*90 2280&70 238Oi90 2520&80 222oi90 2330*70

L 2 2

3

3

3

968 1440 2028 2031

2140,SO 2160*70 198Ok80 2030+70 22OOk80 226Ok80 21OOk90 23OOk70 2090&60 2110&80 2060,70 2120+70 2060+60 2170&70 2000*70 1990*70 237Ok70 2130+60 2170*70 2120&70 1900+60 2040*70 215Ok70 BOO& 100

4

2034 2035 2036 2564 256X 2571 2573 2969 2970 2972 2973 2974 2975 3027 3733 3743 3899 3901 4337 4366

4

4

4 4

4

4

Danebury Full details of the archaeology and radiocarbon dating of Danebury are given in C’unliffe (1984) and are not included here. Briefly however, Danebury is an iron-age hillfort which was excavated over a period of several years and yielded many samples suitable for radiocarbon dating. The radiocarbon results analysed here are from samples taken from contexts closely associated with four ceramic phases. There are a total of 60 acceptable radiocarbon results available, spread fairly evenly throughout the four phases. For ease of exposition. we will call the phases 1, 2. 3 and 4; for full details of the associated pottery sequences see Cunliffe (1984). In Table 3 are given details of the radiocarbon results, as published in Cunliffe (1984), indicating their phase associations (here we reproduce the radiocarbon results in full as some typographical errors appear in those given in Naylor 8: Smith (198X)). Figure 3 shows the same radiocarbon results alongside the corresponding part of the calibration curve.

506 c‘. E. BIJC‘K ET /IL>.

2500 - x 0

X

2400 - . x q

x80

I 1 I I I 0 2600 2400 2200 2000 1800

Calibrated date BP

Dctailed statistical discussion of the radiocarbon results from Danebury have been given by Orton ( 1983) and Naylor & Smith ( 1988). Orton considered estimating the beginnings and endings of the four ceramic phases on the radiocarbon scale. Naylor & Smith used a Baycsian approach to estimating beginning and ending dates for the four phases on the calendar scale, but some problems arose in communication between the experts involved in the analysis. This led to the statisticians adopting what are now considered as outdated calibration curves (the high precision curves were not available when the analyses were undertaken). Another problem was that the offset for dates RP was taken as 1983 AU and not 1950 AL). the value used by the international radiocarbon community. A re-analysis is now particularly timely both to facilitate communication between statisticians and archaeologists and because of the recent innovations in statistical methods based on the Gibbs sampler.

We must emphasize that the primary purpose of our re-analysis is to illustrate the power and capabilities of the Bayesian methodology. We have adopted the same archaeological assignment ofsherds (and their associated radiocarbon dates) to phases as Cunliffe (1984) and Orton (1983). In doing so we have ignored a possible problem with the Danebury example that some of the material dated may be intrusive into older contexts, while in the later phases residual material ofearlier date may have been sampled in a few cases. To deal with this would involve a complete reassessment of the archaeological evidence. which is beyond the scope of this paper.

We initially adopt the same basic model for the interpretation of the Dancbury data as that of both Orton and Naylot- & Smith. The phases arc assumed, on the evidence of the

C‘ALIBRATION OF RADIOCARBON RESULTS 507

ceramic typology. to be abutting so that if u, is the calendar date of the start of phase 1 and U, is the calendar date of the end of phase 1, then the calendar date of the start of phase 2 is ,tiso CI~. Let c(, be the calendar date of the end of phase 2, u4 be the calendar date of the end of phase 3 and u, be the calendar date of the end of phase 4. Then on the basis of the :trchaeological cvidcnce:

u, > u, > u, > a4 > a5 > 0.

end the prior density for u,.u,,u,.uq and u5 is taken as

p(u, +uya,.*,) = c u,>u,>uj>u4>u~>o 0 otherwise.

vvherc c is a constant. Given that we have no other archaeological evidence, we assume that if the ith sample is

associated with phase,j then its calendar date, O,,, is uniformly distributed over the interval (U ,u ). In archaeological terms we are assuming a uniform rate of pottery production ov!er’the interval c(, to a ,+,. Mathematically this is expressed as

1

1

PW,, I@, + ,I = a, ~ U,t I a, > “,, 3 u,+ I

0 otherwise.

After some straightforward algebra, we obtain the following conditional distributions

P(Q,,/.~.~,,,, ((174) f (~dLu.0) 1 exp ~ b), - p(@,,)Y

3 ’ -“, u, > q, 3 (I,, ,:

for j= 2.3.3

/‘(u,i\-.H.u,(i#j),o) % (u, , ~ u,) “1 (u, - cl,,,) ‘i,minO,,P, > u, > niaxO1: i i

and

/~(u,~.\-.e.c~,.~~,.u,.u,.o) % (u, u,) -‘Ia minO,, > ui > 0.

where /I, is the number of samples in phase,; (,j= I .2.3.4). The Gibbs sampler methodology for this problem is therefore implemented as follows:

(i) (ii)

(iii)

(iv)

(VI

(vi)

Choose initial values u’j” > III’!“> u(y)> u$I > u’$) for u,. u,, u3. uq and (x5. For each radiocarbon resuit.i, associated with ceramic phasej, sample 0::’ restricted to the range u’:” > Oi:) 3 a;“,‘, . Sampleu(,‘Ifrom theinterval(maxel,.r,)withprobabilityproportional to(u, -u$“‘))“s.

For ,j= 2,3,4. sample u, (I) from’ the interval (maxfI,,.minO,,~ ,) with probability pt-o- i i

portional to(uj”, -u,))“~ I(u,-u)y,)p”~. Sampleu’j’from theinterval(O,min8,,)withprobabilityproportional to(cc’j’-(I~) m”J.

i Repeat(iv)and(v)with thcsuperscript (0)replaced by( I),( 1)replaced by(2)ctcuntil I’ iterations have been completed. The whole procedure is replicated 111 times.

508 C. E. RUCK ET AL.

2700 2500 2300 2100 I900 0.2

Q2 2, 0.1 -

t - 0 0.0 I I I I I I g 2700 2500 2300 2100 1900

e 0.2 a a3 L 0.1 - 0 0.0 / I I I 1 I i i i

t 2700 2500 2300 2100 1900

a” 0.2 a4

0-I - 0.0 1 I I I I I I

2700 2500 2300 2100 1900 0.2

a5 0-I - 0.0 I I 1 I 1 I

2700 2500 2300 2100 1900

Calrbrated date BP

Figure 4. Posterior probability plots for the calibrated begmning and cndlng dates BP for each of the four phases at Danebury, given the archaeological information that the phases arc abutting and that (within a phase) pottery productjon rates arc uniform. a,(/= 1,2.3.4) IS the beginning date of phase j and ai is the end date of phase 4.

?780 to 23x0 Cal BP 2330 10 2070 Gil HP 28 IO to 2360 Gil HP 2x0 to 181Ocal HP

‘600 to 2360 cal BI’ 22 IO 10 I930 ml HP 2400 to 2 190 Cal HI’ ‘020 to I X50 cal LIP 2180 to2170cal LII’

Adopting this Gibbs sampler methodology using HI= 200 replications and I’ = IO00 iterations, we calculate the posterior density for the dates of each of the q these are plotted in Figure 4. The associated 95% highest posterior probability regions for each of the as in Figure 4 are: u,: 2580 to 2360 cal BP; u,: 2430 to 2300 and 2260 to 2240 cal BP; c(,: 2310 to 2210cal BP; uj: 2240 to 2160cal BP; cl,f 2040 to 1880 cal BP. As with some of the 95% highest posterior probability regions for the events at Runnymede Bridge, we find that the region for u2 at Danebury consists of two parts. This division is caused by the large “wiggle” in the calibration curve at about 2300 cal BP (see Figure 3).

One major archaeological criticism of the Danebury model as used above is: why should the phases be abutting? Arising from this, we might ask: what inferences would be made

C‘ALIBRATION OF RADIOC‘ARBON RESULTS NV

o-0

h ” O-I 2

5 ;1 0.0 2800 2500 2200

g 0.1 a,

t 2 0.0

2800 2500 2200

2300 2000 1700

2300 2000 1700

2300 2000 1700

0.0 I I 2800 2500 2200 2300 2000 1700

Calibrated date BP

Figure 5. Posterior probability plots for the calibrated heginnmg and ending data HP for each of the four phases at Danebury ~hcn the phases arc allowed to owrlap. (1, and 0, are respectively the beginning and ending dates of phase /(/= 1.3.3.4).

t-lrst event u, u4 It

(J / 0.61 1~00 0 78 I ,OO I 00 I ~00 PI 0.00 0.79 woo 0.89 0.19 0.99 (1 ~ 0.67 I Gl O-98 I.00 P: o-00 0.61 0.04 0.9 I (1; 0.98 I a0 PI 0.00 0.94

about the beginning and ending dates of the phases if they were assumed not to be abutting? We might also ask: if we do not make assumptions about the relationship between the phases. can we test whether or not the phases are likely to be abutting?

We now briefly re-analyse the problem using a model that allows the ceramic phases to overlap. This is considered by many archaeologists to be a more realistic model than the abutting one analysed above. Accordingly, let the earliest and latest calendar dates associated with phasej be a, and p, respectively. Suppose that we now make no assumptions about the ordering of the phases except that a,> p,,.j= 1,2,3,4; but we do make the assumption that the rate ofpottery production within any phase is uniform. The phases are assumed to be independent of each other so that

510 C. E. BUCK ETAI..

(‘, a, > B, 0 otherwise

and C, is a constant. Assuming that 8!, is uniformly distributed over the interval (u,. p,) then the conditional density of El,, is porportional to

restricted to the range a, > El,, > p,. The conditional densities of w, and p, are easily shown to be proportional to (a, - fi,))“~ for u, > maxejj and (a, - p,>- ‘J for p, < minO,, respectively.

i i Using the above formulation and the Gibbs sampler methodology the data was re-

analysed with m = 200 and r = 1000. The resulting posterior probability densities of the cl, and piare given in Figure 5 and the 95% highest probability regions are given in Table 4. It is clear from this re-analysis that the calibrated radiocarbon results do not in themselves lead us to believe that the ceramic phases are abutting in time. In fact, the suggestion is that although the phases show a clear progression through time (with the start of each phase falling before the start of the “next” and the end of each phase falling before the end of the “next”) there is considerable overlap in pottery production from the different ceramic phases.

Moreover, from this re-analysis of the data using this second model, we can make some assessment of the relationship between events. It is straightforward to calculate whether the beginning or ending date of phase i precedes the beginning or ending date of phase j(i#tj). The probabilities for these relationships are given in Table 5. For example the probability that phase 2 starts before phase 3 starts is 0.67 and the probability that phase 2 ends before phase 3 ends is 0.62. However the probability that phase 2 ends before phase 3 begins is zero (to two decimal places) thus strongly suggesting that the two phases overlap, but that the start and end dates for phase 2 are both earlier than the respective dates for phase 3.

Similarly, if we consider the other phases there is very strong evidence, based on the radiocarbon results alone, that the phases are overlapping rather than abutting. The calibration ofthe radiocarbon results associated with the pottery phases at Danebury is a clear example of a situation in which combining archaeological and radiocarbon information within the analysis may result in substantially different conclusions being drawn from the data than if the latter were analysed in isolation from the substantive knowledge of context.

Conclusions Applying the Bayesian methodology to the calibration and interpretation of radiocarbon results permits the inclusion of a much wider range of archaeological information than conventional methods. However, to achieve this synthesis, the archaeological information has to be represented (via the prior distribution in Bayes’ theorem) in an appropriate manner capturing the archaeologist’s assumptions.

Having observed the effect of including and not including archaeological information when calibrating the radiocarbon results from Danebury, it is clear that if wc are to be able to make the most useful inferences the data from both archaeological and radiocarbon sources must be of high quality and integrity. Uncertainty about the nature of the archaeological information can be accounted for in the model or by comparing the results using several different models. Uncertainty about the quality of the radiocarbon results is less easily allowed for in the statistical analysis. However. increased accuracy and


precision over the last few years means that the types of statistical analysis advocated here can now really come into their own. The computations involved are straightforward to implement, but may require considerable computer time. For example, on a SUN SPARC station typical runs for the Runnymede Bridge and Danebury examples took about 5 min and 6 h respectively.

Informed readers will be aware that the calibration curve is nof known precisely, but that its knots are known subject to some error term. This error can easily be incorporated into the Gibbs sampler methodology by simulating, for each replication. each knot from its distribution and working with that realization of the curve for the complete run of r

iterations. The results are then averaged over the m replications. Experience with the illustrative examples described in this paper suggests that this makes only a very slight difference to the marginal densities. It should be noted, however, that with increased precision of the radiocarbon results provided by the laboratories, the effect of using such simulated calibration curves will be more marked.

Acknowledgements The authors are grateful to Stuart Needham for making available to us the archaeological information and radiocarbon results from Runnymede Bridge. We also wish to thank David Stephens for initial discussions about the application of the Gibbs sampler methodology to this problem and James Kenworthy for invaluable comments on an earlier draft of the paper. The first named author is supported by the Complex Stochastic Systems Initiative of the SERC Mathematics Committee.

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calibration of radiocarbon results pertaining to related archaeological events

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