The Statistician (2000)
49, Part 3, pp. 339±354
Bayesian modelling of prehistoric corbelled domes
Y. Fan and S. P. Brooks
University of Bristol, UK
[Received March 1999. Final revision April 2000]
Summary. The ®eld of archaeology provides a rich source of complex, non-standard problems
ideally suited to Bayesian inference. We discuss the application of Bayesian methodology to
prehistoric corbelled tomb data collected from a variety of sites around Europe. We show how the
corresponding analyses may be carried out with the aid of reversible jump Markov chain Monte
Carlo simulation techniques and, by calculating posterior model probabilities, we show how to distinguish between competing models. In particular, we discuss how earlier analyses of tomb data by
Cavanagh and Laxton and by Buck and co-workers, where structural changes are anticipated in the
shape of the tomb at different depths, can be extended and improved by considering a wider range
of models. We also discuss the extent to which these analyses may be useful in addressing
questions concerning the origin of tomb building technologies, particularly in distinguishing between
corbelled domes built by different civilizations, as well as the processes involved in their construction.
Keywords: Bayesian model selection; Changepoint models; Gibbs sampler; Log-linear models;
Markov chain Monte Carlo methods; Metropolis±Hastings sampling; Reversible jump Markov chain
Monte Carlo simulation
1. Introduction
The technique of corbelling, a method of spanning or roo®ng spaces with blocks of stone,
balanced on top of each other (Fig. 1), was widely used in prehistory. Surviving examples of its
use include the corbel-vaulted tombs of Brittany, the Bronze Age nuraghi of Sardinia and the
Mycenaean and late Minoan tholoi of Greece and Crete respectively. Other examples of corbelling
may be found in passages and chambers within certain pyramids of Egypt and in prehistoric tombs
on the Orkneys and in Ireland. It was only with the invention of the true dome, enabling much
larger spaces to be spanned, that the techniques of corbelling became less widely used, though
examples of relatively recent origin still exist in southern France, Ireland and Italy.
In recent years, interest in these corbelled domes has led to mathematical models being
developed to investigate how they were constructed, and to compare them between different
civilizations. In particular Cavanagh and Laxton (1981, 1982, 1985) proposed a simple model to
describe the shape of these tombs. The data are in the form of pairs (d i , ri ) where d i denotes the
depth below the apex of the tomb with corresponding radius ri at that depth. Cavanagh and Laxton
proposed a least squares method for ®tting models, to data above the lintel of the tomb, of the form
log(ri ) ˆ log(á) ‡ â log(d i ‡ ä) ‡ E i :
(1)
Thus, the underlying relationship between the radius and depth is that the radius is equal to á
Address for correspondence: S. P. Brooks, Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK.
E-mail: [email protected]
& 2000 Royal Statistical Society
0039±0526/00/49339
340
Y. Fan and S. P. Brooks
Fig. 1. Idealized tomb shape, indicating the various model parameters
times the depth raised to the power â. Clearly á therefore has the interpretation of a scale
parameter and â a shape parameter. Here, ä denotes the distance from the true apex of the dome to
the beginning of measurement for d, since in some cases the dome is truncated below its true apex
by a large `capping' stone; see Fig. 1. The E i are assumed to be independent and identically
distributed normal errors with variance ó 2 . Tombs may be compared by looking at the estimated
shape parameter â from each.
The idea behind the concentration by Cavanagh and Laxton on data above the lintel is that they
believed that there was a fundamental change in the shape of the tomb at this point. To investigate
this possibility, Buck et al. (1993) extended Cavanagh and Laxton's basic model to include a
single changepoint so that the full range of data is described by the model
0 , d i , ã1 ,
log(á1 ) ‡ â1 log(d i ‡ ä1 ) ‡ E i
(2)
log(ri ) ˆ
log(á2 ) ‡ â2 log(d i ‡ ä2 ) ‡ E i
ã1 , d i
where ã1 denotes the depth of the changepoint.
Physical considerations dictate that the curvature of the upper part of the dome must be greater
than that of the lower part, so that â1 . â2 . 0. Furthermore, since the curve must be continuous at
d ˆ ã1 we have the constraint that
á1 (ã1 ‡ ä1 )â1 ˆ á2 (ã1 ‡ ä1 )â2 :
We incorporate this constraint by expressing â1 in terms of the other parameters, so that
â1 ˆ â2 ÿ
log(á1 =á2 )
:
log(ã1 ‡ ä1 )
Both of these models have a realistic physical interpretation. However, there is a far wider
range of equally plausible models, each of which tells us something about the process of the
construction of these domes. In this paper we establish a class of plausible models for these domes
on the basis of exploratory analysis of the data and on the examination of the physical processes
involved in the dome's construction. We carry out a Bayesian analysis using reversible jump
Markov chain Monte Carlo (RJMCMC) simulation techniques to obtain both posterior parameter
estimates and posterior model probabilities which may be used either to discriminate between
Corbelled Domes
341
models or to construct model-averaged posterior inference. Finally, we examine a variety of data
collected from different periods and areas, to illustrate the utility of our approach for comparing
buildings of different origin.
2. Data, models and notation
We begin by focusing on the data provided by Buck et al. (1993), collected from the late Minoan
tholos tomb at Stylos, Crete. The data are provided in Table 1.
Fig. 2 plots log(d i ) versus log(ri ) for the Stylos data of Table 1. By eye, we can see a roughly
log-linear relationship between the depth d and radius r of the tombs with a possible changepoint
at around log(d i ) ˆ 0:8. At larger depths (at around log(d i ) ˆ 1:3), it appears that the tomb wall
becomes vertical, suggesting that we might also wish to add a constant term to the model.
Re-examining the model proposed by Buck et al. (1993) there seems no reason to suggest that
we should have the same values of ä for the two tomb sections of different curvatures, and thus we
might introduce a second ä-term so that we have one corresponding to each of the two curves.
Thus, we obtain a new model given by
8
0 < d i , ã1 ,
< log(á1 ) ‡ â1 log(d i ‡ ä1 ) ‡ E i
ã1 < d i , ã2 ,
(3)
log(ri ) ˆ log(á2 ) ‡ â2 log(d i ‡ ä2 ) ‡ E i
:
log(c) ‡ E i
ã2 < d i
with continuity constraints
á1 (ã1 ‡ ä1 )â1 ˆ á2 (ã1 ‡ ä2 )â2 ,
á2 (ã2 ‡ ä2 )â2 ˆ c,
where c denotes the constant term.
Such a model is entirely plausible in terms of the physical construction of the dome. In fact
there is archaeological evidence to suggest that a third changepoint may also exist, so that there
are four distinct components to the overall structure:
(a) the foundationsÐthis bottom component would typically be very shallow and consist of
only a few layers;
(b) wallingÐthis would typically occur at the bottom of the dome to provide a nearly vertical
wall for the entrance to the structure and would therefore end somewhere around the top of
the lintel or relieving triangle if they exist;
(c) the dome properÐthis component would usually make up the majority of the dome and
Table 1. Depths (d i ) and corresponding radii (r i ) from the
late Minoan tholos at Stylos, Crete
d i (m)
ri (m)
d i (m)
ri (m)
d i (m)
ri (m)
0.04
0.24
0.44
0.64
0.84
1.04
1.24
1.44
0.40
0.53
0.70
0.90
1.06
1.16
1.26
1.36
1.64
1.84
2.04
2.24
2.44{
2.64
2.84
3.04
1.47
1.62
1.67
1.68
1.77
1.82
1.89
1.96
3.24
3.44
3.64
3.84
4.04
4.24
4.44
4.64
2.00
2.05
2.10
2.10
2.14
2.13
2.15
2.14
{Depth of the lintel.
342
Y. Fan and S. P. Brooks
Fig. 2. Plot of the data from Table 1
would usually stretch from the lintel to somewhere close to the apex of the dome;
(d) the capÐthis component would normally exist at the very top of the dome and would have
enabled the builders to make last minute adjustments to the height of the dome.
Thus there is a reasonably large general class of plausible models, containing up to three
changepoints, allowing for a constant slope at the bottom of the dome and allowing for different
ä-values for each component. For comparison with earlier results we shall also consider the case
where all components take the same ä-value. Thus we adopt the notational triple A=B=C for the
models, where A (ˆ 0, . . ., 3) denotes the number of changepoints, B (same S or different D)
relates to the ä-terms and C (ˆ 0, 1) denotes the number of constant components at the bottom of
the dome. For example, the original model of Cavanagh and Laxton given in expression (1) is
denoted by 0=S=0, the model of Buck et al. (1993) given in expression (2) is denoted by 1=S=0
and the model described above and given in expression (3) is denoted by 2=D=1. Thus, we have 12
distinct plausible models (since 0=S=0 0=D=0, 1=S=1 1=D=1 and both 0=D=1 and 0=S=1 are
prohibited) for comparison.
3. The Bayesian approach
The Bayesian approach to statistical modelling uses probability as a means to quantify the beliefs
of the observer about the model parameters, given the data observed. Given a particular model the
approach involves choosing a prior distribution, which re¯ects the observer's beliefs about what
values the model parameters might take before having seen the data, and then updating these
beliefs on the basis of the data observed. The posterior distribution, which is proportional to the
product of the prior distribution and the likelihood function, represents our beliefs having
observed the data. An excellent introduction to the ideas and application of Bayesian techniques
in archaeology is provided by Buck et al. (1996).
Given a particular model M i for the data x, with parameters èi , we may then assume a prior
p(èi ) for those model parameters. Thus, the corresponding posterior distribution is given by
ð(èi jx) / p(èi ) Li (èi ; x),
Corbelled Domes
343
where Li (èi ; x) denotes the likelihood associated with the data under model i.
Suppose that we wish to make inferences from some target distribution, ð(è), è 2 È R k ,
which need be known only up to some multiplicative constant. In our context, ð is the posterior
distribution and è the vector of model parameters. We construct a Markov chain with state space
È and whose stationary (or invariant) distribution is ð(è), as discussed in Smith and Roberts
(1993), for example. Then, if we run the chain for suf®ciently long, simulated values from the
chain may be treated as a sample from the target distribution and used as a basis for summarizing
important features of ð.
In simulating our Markov chain, we update the model parameters one at a time by sampling
from the full conditional for that parameter given all the others. However, the convergence rate of
the chain may sometimes be increased by updating blocks of parameters simultaneously, as
described by Roberts and Sahu (1997). We cycle through the parameters, updating each in turn,
and denote a complete cycle as a single MCMC iteration, moving the chain from state è t to è t‡1 ,
say. Given that we wish to update èi , we draw a proposed new value ö from some arbitrary
proposal density qi (èi , öjè(i) ), where è(i) ˆ fè1 , . . ., èiÿ1 , èi‡1 , . . ., è k g. We accept the newly
proposed value ö with probability
(
)
ð(öjè(i) ) qi (ö, èi jè(i) )
:
(4)
á(èi , öjè(i) ) ˆ min 1,
ð(èi jè(i) ) qi (èi , öjè(i) )
If the new value is rejected, we set è it‡1 ˆ è it .
The choice of proposal distribution is fairly arbitrary, though it is clear from equation (4) that,
if qi (ö, èi jè(i) ) ˆ ð(èi jè(i) ), then we automatically accept every proposed new value. Updates of
this sort are a special case of the general Metropolis±Hastings update, known as a Gibbs update.
In practice, Gibbs updates are the most ef®cient if the corresponding conditional distribution is
simple to sample from directly. See Brooks (1998) for further discussion on this topic.
Posterior inference may then be obtained by calculating the posterior marginal distributions for
each of the parameters, for example, or by calculating point estimates for parameters of interest
which, for any particular loss function, minimize the posterior expected loss. Such estimates
are known as Bayes estimates and are discussed in Bernardo and Smith (1994), for example.
Commonly, the quadratic loss function is taken, so that the Bayes estimate is simply the posterior
mean.
The Bayesian paradigm also provides a very natural framework for considering several models
simultaneously, either assigning probabilities to the individual models or averaging predictive
inference over the full set. Various computational techniques have been proposed for estimating
the posterior model probabilities; see Clyde (1999) and Gamerman (1997), for example. In
particular, the RJMCMC algorithm proposed by Green (1995) is ideally suited to deal with
Bayesian model determination problems.
Given a set of models M 1 , . . ., M k say, which a priori we are willing to consider as realistic
alternatives for describing a particular data set, it is possible to derive probabilities associated with
each model, which may then be used to discriminate between them. We begin by assigning a prior
probability to each model. Commonly, these prior probabilities may be equal, representing the
assumption that each model is equally likely. Alternatively, the prior probabilities may be some
function of the number of parameters in each model, so that models with large numbers of
parameters are penalized in some way. The choice of prior is entirely at the analyst's discretion,
but it should be based on all available information about the problem at hand before the data were
collected. For illustrative purposes, we shall associate probability pi with model M i and, for the
remainder of this paper, assume that all models under consideration are equally likely a priori.
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Y. Fan and S. P. Brooks
As before the priors for the models and their corresponding parameters can be combined with
the likelihood to obtain a full joint posterior distribution over both the model and the parameter
space. RJMCMC simulation allows us to sample from this joint posterior distribution, thereby
providing estimates of model probabilities within the MCMC simulation itself by simply observing the number of times that the chain visits each distinct model. These might be best described as
relative posterior probabilities. If for example there were models which were not possible to
consider during the simulation, e.g. a family of distributions not allowed for within the algorithm,
and with non-zero posterior probability, then their introduction into the analysis would reduce the
corresponding posterior probabilities of the existing models. However, their relative values would
remain unchanged. Another interpretation of these probabilities might be as conditional on the
range of models being considered.
Simply put, RJMCMC simulation extends the basic Metropolis±Hastings algorithm to general
state spaces, so that the target distribution ð becomes a general measure, rather than a density, and
the Metropolis±Hastings proposal density is replaced by a proposal kernel.
Green (1995) provides a `template' for reversible jump moves. Suppose that a move of type m
from some countable family is proposed, from a point è to another point ø in a higher dimensional
space. This will very often be implemented by drawing a vector of continuous random variables u,
independent of è, and setting ø to be a deterministic and invertible function ø(è, u). The reverse
of the move (from ø to è) can be accomplished by using the inverse transformation, so that, in
this direction, the proposal is deterministic. Thus, if ð(dè) qm (è, dø) has a ®nite density with
respect to some symmetric measure, then we can generate (m, ø) f m (è, ø) and we accept this
move with probability min(1, A), where
f m (ø, è) rm (ø) @ø ,
Aˆ
f (è, ø) r (è) q(u) @(è, u) m
m
rm (è) is the probability of choosing move type m when in state è and q(u) is the density function
of u. The ®nal term in the ratio above is a Jacobian arising from the change of variable from (è, u)
to ø.
Having discussed the general approach to Bayesian inference and outlined the simulation
techniques that are involved, we next discuss the implementation of these methods for the problem
at hand.
4. Implementation
The implementation can be broken down into two distinct areas. The ®rst is the MCMC simulation
involved in updating the parameters conditionally on a particular model. The second concerns the
jumps between models as we add or delete changepoints and merge or split ä-parameters. Before
we discuss the two different move types involved in the simulation, we must ®rst decide on
suitable prior distributions for the parameters to de®ne the associated posterior distribution for
simulation.
Following Buck et al. (1993), we take N (0, 1) priors for log(c) and for the log(á i ) (these are
rather vague since values with a modulus of greater than 1 are rarely observed), N (1, 1) priors for
the â-parameters and U (0, id n =2k) priors for ä i > ä j for i > j and k denotes the number of
components. We take U (d 1 , d nÿ1 ) priors for the changepoints, constrained so that ã i . ã j , i > j.
Finally, we take a vague Ã(1, 0:01) prior for the inverse of ó 2 , the error variance. Of course, these
priors are chosen somewhat subjectively, to re¯ect our beliefs about the model parameters before
observing any data.
Here, we have adopted priors which are reasonably vague, yet consistent with the underlying
Corbelled Domes
345
processes which we believe are involved in constructing the tombs. More informative priors might
have been adopted; for example there is some archaeological evidence that in Greek tholoi the
walling below the door lintel is constrained within a rock cylinder whereas from thereon upwards
there is no such constraint. This would suggest a prior that places a changepoint fairly close to the
lintel. It is always important to incorporate any such knowledge where practicable, and also to
carry out sensitivity analyses to determine the in¯uence that any such assumptions may have on
the resulting inference. In practice and for the example discussed in this paper, we have found that
the posterior inference is broadly insensitive to the choice of priors within a reasonable range.
4.1. Within-model moves
In general, the posterior conditional distributions for the model parameters are all non-standard
(i.e. they are not of familiar form nor easy to sample from directly), though there are some
exceptions. For example, under model 1=S=0, Buck et al. (1993) derived the form of the
conditionals for the log(á i )- and â1 -parameters. However, since the simulation involves movement
between many different models it is most ef®cient to use Metropolis±Hastings updates for all
parameters. The one exception is that the posterior conditional distribution for ó 2 is always an
inverse gamma distribution, with parameters n=2 ‡ 1 and S=2 ‡ 0:01 under our chosen priors,
and where
P
flog(ri ) ÿ log(á1 ) ÿ â1 log(d i ‡ ä1 )g2
Sˆ
d i <ã1
‡
P
ã1 , d i <ã2
flog(ri ) ÿ log(á2 ) ÿ â2 log(d i ‡ ä2 )g2 ‡
P
d i . ã2
flog(ri ) ÿ log(c)g2
under model 2=D=1, for example. Similar expressions are easily available for S under each
separate model.
To update the remaining parameters, we use random walk Metropolis updates with the
following proposals:
á9i U (á i ÿ 0:5, á i ‡ 0:5),
â9i U ( â i ÿ 0:5, â i ‡ 0:5),
ä9i U fmax(0, ä iÿ1 , ä i ÿ 0:1), min(id n =2k, ä i‡1 , ä i ‡ 0:1)g,
ã9i U fmax(d 1 , ã iÿ1 , ã i ÿ 0:5), min(d nÿ1 , ã i‡1 , ã i ‡ 0:5)g:
The corresponding acceptance probabilities are then individually derived under each model from
equations (4).
4.2. Between-model moves
Here we distinguish between two separate between-model move types. The ®rst either adds or
removes a changepoint from the model and the second allows us to move between models with
one and multiple ä-values.
4.2.1. Adding and deleting changepoints
There are two separate situations to consider, adding or deleting a changepoint, depending on
whether or not the additional component to the model is a curve or a constant. Thus, we consider
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Y. Fan and S. P. Brooks
two types of move in this particular problem; one type is to add or delete curves and the other is to
add or delete the constant term associated with the modelling of the bottom part of the tomb. In
both of these move types, the number of changepoints is altered; therefore RJMCMC updates are
needed to accommodate these dimension jumping moves.
4.2.1.1. Add or delete curve move. Suppose that our current model consists of k components,
1 < k < 4. For simplicity, we assume that the model consists only of curved components. Adding
or deleting a curve to a model with a constant term proceeds similarly except that the curve is
added to just above the constant component. In practice, both types of model need to be
considered. If we let á, â, ä and ã denote original values and let á9, â9, ä9 and ã9 denote proposed
new values for the model parameters, then we adopt the following proposal mechanism. Firstly,
we generate three random proposal variables: u1 N (0, 0:5), u2 N (0, 0:1) and u3 U [ã k , d nÿ1 ]. We then set
á91 ˆ á1 , . . ., á9kÿ1 ˆ á kÿ1 ,
á9k ˆ á k ÿ u1
and
á9k ‡1 ˆ á k ‡ u1 ,
and
ä9k ‡1 ˆ ä k ‡ u2 ,
â91 ˆ â0 ,
ä91 ˆ ä1 , . . ., ä9kÿ1 ˆ ä kÿ1 ,
ä9k ˆ ä k ÿ u2
ã91 ˆ ã1 , . . ., ã9k ˆ ã k
and
ã9k ‡1 ˆ u3 :
Note that the ä does not change if all the ä i are the same, i.e. when we are in a model of the form
A=S=C for any A and C. The Jacobian term of this transformation is 4; therefore the acceptance
probability reduces to min(1, A), where
Aˆ
4ð m9 (á9, â9, ä9, ã9, ó 2 ; x) p(k ‡ 1, k)
ð m (á, â, ä, ã, ó 2 ; x) p(k, k ‡ 1) q(u1 , u2 , u3 )
where p(k, k9) denotes the probability of proposing to move to model m9 with k9 changepoints
from model m with only k, ð m denotes the posterior density for the parameters under model m
and q(u1 , u2 , u3 ) denotes the joint density of the three proposal variables. Note that the posterior
density term comprises the product of a likelihood and the relevant prior terms. As we shall be
jumping between models, it is essential that each of these prior terms is properly normalized, since
those in the numerator will no longer cancel with those in the denominator.
The reverse move (deleting a curve) is then deterministic and the auxiliary variables u1 , u2 and
u3 are derived by inverting the above transformations. The move is then accepted with probability
min(1, 1=A).
4.2.1.2. Add or delete constant move. The add a constant move involves simply generating a
new changepoint,
ã9k ˆ u
where u U [ã kÿ1 , d nÿ1 ], and then setting
ä9k‡1 ˆ 0,
â9k ‡1 ˆ 0,
á9k ‡1 ˆ á9k ‡ â9k log(ã9k ‡ ä9k ):
All other parameters remain ®xed. The Jacobian term for this transformation is 1 and so the
acceptance probability for this move becomes min(1, A), where
Corbelled Domes
Aˆ
347
ð m9 (á9, â9, ä9, ã9, ó ; x) p(k ‡ 1, k)
ð m (á, â, ä, ã, ó 2 ; x) p(k, k ‡ 1) q(u)
2
with p(k, k9) ˆ 1 de®ned in a manner similar to that above. Here the reverse move (deleting a
constant term) again has acceptance probability min(1, 1=A).
4.3. Splitting and merging ä-values
Splitting ä-values involves changes only to the ä-parameter; all other parameter are unaffected.
Although this move involves a change in dimensions, the number of components remains unchanged.
For a k . 1 component model, we split the ä-values by generating proposal variables
u1 , . . ., u kÿ1 , where
u1 U [0, 2],
u2 U [u1 , 2], . . .,
u kÿ1 U [u kÿ2 , 2]:
We then set
ä91 ˆ ä ÿ u1 ,
ä92 ˆ ä ‡ u1 ,
ä93 ˆ ä ‡ u2 , . . .,
ä9k ˆ ä ‡ u kÿ1 :
The acceptance probability for this move is min(1, A) where
Aˆ
ð m9 (á, â, ä9, ã, ó 2 ; x) p(ä( k) , ä)jJ j
ð m (á, â, ä, ã, ó 2 ; x) p(ä, ä( k) ) q(u1 , u2 , . . ., u kÿ1 )
where the Jacobian term jJ j ˆ 2 and p(ä( k) , ä) ˆ p(ä, ä( k) ) ˆ 1 denotes the probability of moving
from the model with different ä-values to the model with the same ä-values and vice versa. The
term
q(u1 , u2 , . . ., u kÿ1 ) ˆ q(u1 ) q(u2 ju1 ) q(u3 ju2 ) . . . q(u kÿ1 ju kÿ2 )
ˆ
1 1
1
1
...
:
2 2 ÿ u1 2 ÿ u2
2 ÿ u kÿ2
The reverse move, in which we merge ä-values, is then achieved by setting
ä ˆ (ä91 ‡ ä92 )=2
and the ui are obtained deterministically by inverting the transformations used in the split ä move.
The acceptance probability is min(1, 1=A).
Having described the necessary steps within the MCMC simulation algorithm, we now present
the analysis of the Stylos data.
5. Results for the Stylos data
We run our MCMC chain for 15 3 106 iterations, discarding the ®rst half of the simulation as
burn-in. Very large numbers of iterations are needed because autocorrelation plots of the sampler
output indicate high serial correlations and thus slow mixing. This high correlation is due to the
imposition of the continuity constraints within the model. In addition, we store only one iteration
in every 3000 to minimize expense in terms of computer storage. Thus, we base our inference on a
resulting (approximately independent) sample of 2500 observations from the full joint posterior.
We recall that the posterior mean minimizes the expected loss under the quadratic loss function.
Other loss functions could be used to obtain point estimates. For example the posterior median is
the Bayes estimate under the absolute error loss function. However, in the examples discussed
348
Y. Fan and S. P. Brooks
here, the posterior medians were all very close to the posterior means, and similar inference is
obtained.
Table 2 provides the posterior model probabilities for the models under consideration (i.e. up to
and including three changepoints). From this, we can see that model 3=D=1 is clearly identi®ed as
being a posteriori most likely, with three changepoints, different ä-values for each component and
a vertical component at the bottom of the dome. There is also some posterior support for model
2=D=0 and for models 2=D=1 and 2=S=1. All these models are `neighbours' in the sense that you
can move from 3=D=1 to 2=D=0 by the deletion of the constant component, from 3=D=1 to 2=D=1
by the deletion of a curved component and from 2=D=1 to 2=S=1 by merging the separate ävalues. We note that the top three models have different ä-terms, suggesting that Buck et al.
(1992) were too restrictive in adopting the assumption that each curve had the same ä-value. In
fact 76% of the posterior mass is placed on models which have different ä-values for each
component and their model (1=S=0) has a posterior model probability of only 3.1%. Similarly, the
model originally proposed by Cavanagh and Laxton (1981) (0=S=0) has no posterior support at
all. This suggests that the broader class of models considered in this paper provides a far better
description of the dome while retaining the interpretability in terms of the methods used in its
construction.
There is some support for a vertical region at the bottom of the dome in that three of the top
four models have a constant term and 60% of the posterior mass is placed on models with this
term. Finally, we can see that the posterior probabilities of their being no, one, two or three
changepoints are approximately 0%, 8%, 46% and 47% respectively. From an ad hoc inspection
of Fig. 2 these probabilities seem quite plausible.
Table 3 provides the parameter estimates corresponding to the four a posteriori most probable
models, together with the corresponding 95% highest posterior density interval (HPDI) estimates.
We can see that there is some evidence for two changepoints at depths of around 1.3 m and 3.7 m
and the possibility of a third at around a depth of 2.7 m. This is entirely consistent with an ad hoc
interpretation of Fig. 2.
Interpreting the results of Tables 2 and 3, we might conclude that the dome was divided into
three or four distinct layers. The bottom layer is essentially vertical with a radius of 0.76 m and
rises to a height of about 0.8 m. The second layer (which might comprise two separate sections)
rises to a height of roughly 3.4 m and, though there is some uncertainty, has a â-value of
somewhere between 0.5 and 0.7. The ®nal layer rises for another 1.2±1.6 m and has a â-value of
around 0.9. The dome was capped approximately 0.4 m below the true apex of the top layer. The
two a posteriori most probable models both support the hypothesis of a changepoint at the lintel
in that they have a ã-parameter with a corresponding HPDI which encloses that value. Indeed,
model 3=D=1 even suggests a changepoint somewhere around 2.7 m. Given that depths are
recorded at intervals of 20 cm this not inconsistent with a changepoint at the lintel depth. Thus,
Table 2. Posterior model probabilities for the Stylos data, using the A=B=C notation{
Probabilities for the following values of A and B ˆ D or B ˆ S:
A
0
B
C
0
1
1
2
3
D
S
D
S
D
S
D
S
Ð
Ð
0.000
Ð
0.044
Ð
0.031
0.001
0.202
0.117
0.035
0.104
0.071
0.326
0.019
0.052
{0=D=1 and 0=S=1 are not allowed and 0=D=0 0=S=0 and 1=D=1 1=S=1. The a posteriori most
probable model is given in bold.
Table 3. Summary of parameter estimates corresponding to the top four models for the Stylos data
Parameter
Estimates for the following models:
3=D=1
Model probability
2=D=1
2=S=1
Mean
95% HPDI
Mean
95% HPDI
Mean
95% HPDI
Mean
95% HPDI
ÿ0.21
ÿ0.24
0.00
0.76
0.96
0.71
0.47
0.42
0.83
1.36
1.22
2.69
3.89
0.002
(ÿ0.38, ÿ0.01)
(ÿ0.61, 0.05)
(ÿ0.49, 0.61)
(0.72, 0.80)
(0.67, 1.23)
(0.46, 0.99)
(0.05, 0.74)
(0.20, 0.58)
(0.41, 1.13)
(0.81, 1.71)
(0.69, 1.94)
(1.54, 3.82)
(3.37, 4.37)
(0.001, 0.003)
ÿ0.18
ÿ0.16
0.25
Ð
0.91
0.63
0.30
0.39
0.81
1.39
1.38
3.12
Ð
0.002
(ÿ0.38, ÿ0.01)
(ÿ0.48, 0.15)
(ÿ0.20, 0.70)
Ð
(0.64, 1.21)
(0.39, 0.88)
(0.02, 0.54)
(0.18, 0.57)
(0.36, 1.20)
(0.83, 1.83)
(0.77, 2.11)
(1.88, 4.05)
Ð
(0.001, 0.003)
ÿ0.16
ÿ0.14
Ð
0.76
0.87
0.58
Ð
0.37
0.95
Ð
1.55
3.74
Ð
0.002
(ÿ0.48, 0.01)
(ÿ0.54, 0.17)
Ð
(0.72, 0.80)
(0.58, 1.21)
(0.35, 0.80)
Ð
(0.17, 0.62)
(0.38, 1.47)
Ð
(0.85, 2.35)
(3.14, 4.22)
Ð
(0.001, 0.003)
ÿ0.21
0.02
Ð
0.76
0.96
0.52
Ð
0.43
Ð
Ð
1.36
3.71
Ð
0.002
(ÿ0.54, 0.00)
(ÿ0.21, 0.26)
Ð
(0.72, 0.80)
(0.65, 1.39)
(0.32, 0.67)
Ð
(0.18, 0.69)
Ð
Ð
(0.74, 2.10)
(3.24, 4.25)
Ð
(0.001, 0.003)
0.326
0.202
0.117
0.104
Corbelled Domes
log(á1 )
log(á2 )
log(á3 )
log(c)
â1
â2
â3
ä or ä1
ä2
ä3
ã1
ã2
ã3
ó2
2=D=0
349
350
Y. Fan and S. P. Brooks
the hypothesis that the method of construction of this dome might naturally have led to a change
in shape at the lintel is certainly consistent with the data, though this analysis does not really
supply any additional evidence to support this hypothesis.
6. Further analyses
For many decades, historians and archaeologists have had many speculative theories about the
diffusion of the sophisticated building knowledge required to construct corbelled domes. For
example, did the constructors of the nuraghi in Sardinia learn the corbelling technique from the
Greeks or did the construction techniques develop independently in the western and eastern
Mediterranean? Similarly, did the Minoans import craftsmen from the mainland while under
Mycenaean rule? Clearly, it would be useful to have a simple method for distinguishing between
corbelled domes of different shape. In this section, we illustrate how Bayesian model selection can
be easily applied to several sets of data from various sites, and we discuss how comparative
inference might be made.
For illustration, we chose one set of late Minoan tholos data collected from Achladia on Crete
(Cavanagh and Laxton, 1982), one set of Mycenaean data collected (only down to the lintel) from
Dimini (Cavanagh and Laxton, 1981) and one set of nuraghe data collected from Madrone,
Silanus (Cavanagh and Laxton, 1985). These data are provided in Table 4.
As with our previous analysis of the Stylos data, we begin by calculating the posterior model
probabilities for the three new data sets using priors that are similar to those described in Section
4. The posterior model probabilities are given in Table 5.
We can see from Table 5 that each data set leads to a different ordering of the models in terms
of posterior model probability. There are no discernible similarities between the data sets in terms
of these probabilities other than the fact that they all (including the Stylos data) ascribe some
posterior mass to the model 2=D=0. In addition, models 1=D=0 and 1=S=0 have signi®cant
posterior support with all except the Stylos data. Of course, the Dimini dome differs from the
others in that measurements were taken only down to the depth of the lintel. Thus, we might
expect one fewer changepoint for these data, which is exactly what we observe in Table 5. Note
also that the a posteriori most probable model for the Dimini data has two changepoints but does
not include a constant term. In fact only 26% of the posterior mass is placed on models which
include a constant term. This is, of course, because the data are only measured down to the depth
of the lintel and we would usually only expect a vertical layer to the dome to exist below this
point. Thus, the results of Table 5 seem entirely plausible.
Table 6 lists the Bayes estimates for model parameters with the three new data sets calculated,
in each case, only for the a posteriori most probable model.
The results in Table 6 provide somewhat stronger evidence to support the hypothesis of a
changepoint at the lintel than does the analysis of the Stylos data. Obviously, we learn very little
from the analysis of the Dimini data set, but we note that the changepoints at 2.6 m for Achladia
and 4.11 m for the nuraghe data correspond closely to the lintel depths.
The Achladia analysis suggests the presence of either one (with posterior probability 0.366) or
two (with posterior probability 0.483) changepoints, one at the lintel and another at a height of
approximately 0.8 m. The dome consists of a vertical side of roughly 0.7 m to the ®rst changepoint
where it becomes curved (with a â-value of about 0.5) to the lintel. Above the lintel, the curve
changes (with a â-value of about 0.8) and the dome is capped at a depth of between 0.3 and 0.4 m
from the `true' apex of the top layer.
The Dimini dome has only a single changepoint, approximately 1.6 m above the lintel, below
which we observe a curved layer with a â-value of about 0.54 and above which we observe a curve
Corbelled Domes
351
Table 4. Depth (d i ) and corresponding radii (r i ) for the
various corbelled domes in metres
Achladia
Dimini
Nuraghe
d i (m)
ri (m)
d i ( m)
ri (m)
d i (m)
ri (m)
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4{
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
0.45
0.51
0.69
0.86
0.95
1.03
1.10
1.24
1.36
1.44
1.54
1.63
1.72
1.79
1.90
1.91
1.94
1.96
1.96
2.00
1.99
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3.0
3.3
3.6
3.9
4.2
4.5
4.8{
0.99
1.23
1.49
1.62
1.91
2.22
2.49
2.69
2.91
3.05
3.23
3.33
3.48
3.59
3.66
0.12
0.32
0.52
0.72
0.92
1.12
1.32
1.52
1.72
1.92
2.12
2.32
2.52
2.72
2.92
3.12
3.32
3.52
3.72
3.92{
4.32
4.52
4.92
5.12
5.32
5.52
5.72
0.35
0.52
0.61
0.72
0.85
0.96
1.10
1.16
1.28
1.34
1.46
1.49
1.60
1.51
1.70
1.71
1.74
1.77
1.82
1.86
2.02
2.07
2.03
2.05
2.16
2.17
2.23
{Depth of the lintel; there is no lintel for the Dimini data set.
Table 5. Posterior model probabilities for the nuraghe, Dimini and Achladia data, using the A=B=C notation{
C
Data set
Probabilities for the following values of A and B ˆ D or B ˆ S:
0
0
1
Nuraghe
Dimini
Achladia
Nuraghe
Dimini
Achladia
1
2
3
D
S
D
S
D
S
D
S
Ð
Ð
Ð
Ð
Ð
Ð
0.001
0.043
0.004
Ð
Ð
Ð
0.261
0.229
0.156
Ð
Ð
Ð
0.121
0.242
0.105
0.001
0.016
0.105
0.329
0.132
0.120
0.052
0.082
0.176
0.015
0.025
0.027
0.023
0.108
0.160
0.115
0.017
0.013
0.074
0.043
0.098
0.007
0.006
0.007
0.003
0.012
0.031
{0=D=1 and 0=S=1 are not allowed and 0=D=0 0=S=0 and 1=D=1 1=S=1. The a posteriori most probable model is
given in bold.
with a â-value of approximately 0.95. The dome appears to be capped about 0.5 m below its true
apex.
Finally, the nuraghe data appear to arise from a dome having one (with probability 0.384) or
two (with probability 0.419) changepoints, one lying at around the lintel at a depth of approximately 1.9 m from the top of the dome. The dome does not appear to have a vertical component at
352
Y. Fan and S. P. Brooks
Table 6. Summary of results for the three corbelled domes{
Parameter
Estimates for the following data sets:
Achladia
log(á1 )
log(á2 )
log(á3 )
log(c)
â1
â2
â3
ä or ä1
ä2
ä3
ã1
ã2
ó2
Dimini
Nuraghe
Mean
95% HPDI
Mean
95% HPDI
Mean
95% HPDI
ÿ0.34
ÿ0.06
Ð
0.68
0.83
0.51
Ð
0.35
0.90
Ð
2.60
3.45
0.002
(ÿ0.60, ÿ0.14)
(ÿ0.67, 0.60)
Ð
(0.64, 0.73)
(0.67, 1.07)
(0.00, 0.89)
Ð
(0.11, 0.58)
(0.35, 1.34)
Ð
(0.99, 3.27)
(3.00, 3.94)
(0.001, 0.004)
ÿ0.14
0.41
Ð
Ð
0.95
0.54
Ð
0.54
Ð
Ð
3.20
Ð
0.002
(ÿ0.44, 0.16)
(ÿ0.08, 0.97)
Ð
Ð
(0.75, 1.13)
(0.14, 0.82)
Ð
(0.17, 0.86)
Ð
Ð
(2.07, 4.12)
Ð
(0.001, 0.005)
ÿ0.36
ÿ0.36
ÿ0.09
Ð
0.84
0.64
0.44
0.33
1.01
1.73
1.94
4.11
0.002
(ÿ0.72, ÿ0.17)
(ÿ0.70, ÿ0.08)
(ÿ0.61, 0.59)
Ð
(0.61, 1.14)
(0.44, 0.85)
(0.06, 0.68)
(0.09, 0.64)
(0.30, 1.57)
(0.90, 2.32)
(1.19, 2.70)
(2.07, 5.39)
(0.001, 0.003)
{Only the models with highest posterior probability are given.
its base (with probability of only 0.15) but to comprise three curves with â-values of 0.44, 0.64
and 0.84 moving from the bottom to the top. The dome appears to be capped approximately 0.3 m
below its true apex.
Archaeologists have always speculated that the Minoan and Mycenaean tombs had a common
tradition of building techniques. To investigate this, Cavanagh and Laxton (1982) ®tted a simple
regression model (0=S=0) and observed that the â-values appeared most valuable in distinguishing
between domes of different origin. Recall that â denotes the shape of the dome, and á the size.
They could infer that â-values for sets of Mycenaean and Minoan tombs tended to lie in the range
0.65±0.75, though their ®ndings were based only on the analysis of the 0=S=0 model.
The analysis presented here suggests that the Minoan dome at Achladia has two distinct âvalues of about 0.5 and 0.80 and, for the Mycenaean dome at Dimini, values of 0.54 and 0.95, all
of which lie some point beyond the limits suggested by Cavanagh and Laxton (1982). This
apparent discrepancy can be explained by observing that the range obtained by Cavanagh and
Laxton (1982) lies midway between the values observed in our analysis. Essentially, by considering only a single regression line for the entire depth of the dome they obtained a `compromise' âvalue somewhere between the two values that we observe.
Cavanagh and Laxton (1981) also studied the range of â-values associated with nuraghi domes
under the simple linear regression model. They obtained â-values of between 0.42 and 0.60
whereas we observe three distinct values of around 0.8, 0.6 and 0.4. Again this is consistent with
the results of Cavanagh and Laxton (1981) when we take into account the broader range of models
considered in this analysis.
The analyses presented here suggest that the use of â-values to discriminate between tombs of
different construction is not particularly effective when we consider more realistic models of these
domes. Further, it is possible that the conclusions reached by Cavanagh and Laxton (1981, 1982)
were an anomalous consequence of their reliance on a single linear regression model. Even more
disappointing is the fact that the analyses presented in this paper suggests that there is no reliable
way of distinguishing between domes of different origin simply by their shape.
Corbelled Domes
353
7. Discussion
In this paper we have illustrated the Bayesian approach to the statistical modelling of prehistoric
corbelled vaults and have shown how MCMC simulation may be used to facilitate the corresponding analysis. We have also shown how the problem of model selection can be tackled fairly
straightforwardly and we have highlighted the importance of considering a variety of plausible
models when analysing data of this sort. We have shown how this can be done very easily through
the calculation of posterior model probabilities and we illustrated how misleading results may be
obtained if we focus on models which are too simplistic.
In particular, we have examined whether or not it is possible to distinguish between domes of
different origin through a comparison of their shape. Cavanagh and Laxton (1982) suggested that
the â-parameter might be used to distinguish between domes. We found no evidence either to
dispute or to support that claim. However, by considering a range of plausible models rather than
focusing on just one, we showed how these original analyses may have been misleading in this
regard.
The reason that comparisons between domes are inconclusive is essentially the degree of
uncertainty associated with the different parameter values and the â-values in particular. This is a
consequence of the necessarily small amount of data that are available from any one dome. If a
comparison of this sort were of primary interest, then it might be more effective to consider
modelling several similar domes simultaneously to obtain more accurate estimates. For example
we might take 10 separate nuraghi and model them under the assumption that they all have the
same number of components and the same â-values. The parameter á would have to vary between
domes as they would, most likely, each be of a different size. This would provide much more
information on the key parameters of interest and might provide a more informative means of
comparing domes of different origin.
Finally, we note that alternative models may provide a better ®t to the data than those discussed
here. For example, the Stylos data are modelled reasonably well by a cubic regression line. In this
paper we have focused on a restricted class of models which retain a coherent archaeological
interpretation and we either choose the `best' or average over the entire class. The restrictions that
we impose are based on physical construction and archaeological principles (Cavanagh and
Laxton, 1981; Buck et al., 1993) and on the knowledge that the walls of these domes may have
moved over time so that they no longer have exactly the same characteristics as when they were
built. Thus, though other models may provide a better ®t to the data that we observe, we should
restrict our attention only to the archaeologically plausible models discussed in this paper.
Acknowledgements
The authors gratefully acknowledge the Engineering and Physical Sciences Research Council for
their support of the ®rst author under grant GR/L55650 and Dr Caitlin Buck, Dr Bob Laxton, Dr
Nick Fieller, the Joint Editor and three referees for their constructive comments, criticisms and
suggestions, all of which contributed to a greatly improved ®nal manuscript.
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