Geoff K. Nicholls

Radiocarbon dating with temporal order constraints

Bayesian methods are now widely used for analysing radiocarbon dates. We find that the non‐informative priors in use in the literature generate a bias towards wider date ranges which does not in general reflect substantial prior knowledge. We recommend using a prior in which the distribution of the difference between the earliest and latest dates has a uniform distribution. We show how such priors are derived from a simple physical model of the deposition and observation process. We illustrate this in a case‐study, examining the effect that various priors have on the reconstructed dates. Bayes factors are used to help to decide model choice problems.

New radiocarbon calibration software.

We have developed a software utility, DateLab, for conventional radiocarbon age (CRA) calibration and Bayesian analysis of CRAs. The current version has a smaller range of applicability than other similar utilities such as Bcal, Oxcal, and Mexcal. However, it enables analysis of some common types of CRA datesets. The main advantages of DateLab are its high quality sampling algorithm, the possibility of carrying out model comparison and hypothesis testing in a straightforward way, and the unbiased character of the summary statistics on which the analysis depends.

Bayesian image analysis with Markov chain Monte Carlo and coloured continuum triangulation models

It is now possible to carry out Bayesian image segmentation from a continuum parametric model with an unknown number of regions. However, few suitable parametric models exist. We set out to model processes which have realizations that are naturally described by coloured planar triangulations. Triangulations are already used, to represent image structure in machine vision, and in finite element analysis, for domain decomposition. However, no normalizable parametric model, with realizations that are coloured triangulations, has been specified to date. We show how this must be done, and in particular we prove that a normalizable measure on the space of triangulations in the interior of a fixed simple polygon derives from a Poisson point process of vertices. We show how such models may be analysed by using Markov chain Monte Carlo methods and we present two case-studies, including convergence analysis.

Statistical inversion of South Atlantic circulation in an abyssal neutral density layer

This paper introduces a Bayesian inversion approach to estimating steady state ocean circulation and tracer fields. It is based on a quasi-horizontal flow model and a PDE solver for the forward problem of computing solutions to the tracer field advection-diffusion equations. A typical feature of existing ocean circulation inverse methods is a preprocessing stage in which the tracer data are interpolated over a regular grid and the interpolation error is ignored in the subsequent inversion. Our approach only uses interpolated data at those grid points that have neighboring hydrographic stations. By exploiting physically-based models in an integrated fashion, the method provides a statistically unified inversion and tracer field reconstruction with minimal data smoothing. Solving the problem consists of finding information about the circulation and tracer fields in the presence of a number of assumptions (prior information); the resulting posterior probability distribution summarizes what we can know about these fields. We develop a Markov chain Monte Carlo simulation procedure to extract information from the (analytically intractable) posterior distribution of all the parameters in the model; uncertainty about the “solution” is represented by variation in the output of the simulation runs. Our approach is aimed at finding the time-averaged quasi-horizontal flow and tracer fields for an abyssal neutral density layer in the South Atlantic. The collection of oceanographic data during the past century has formed the basis for our understanding of the distribution of properties and the circulation in the ocean. However, a description of the steady flow and associated property fields in the oceanic interior that is dynamically consistent with those collective observations and that accounts for the inherent uncertainty associated with measurements, physical variability and model parameterizations, for example, is still lacking. It may be that the steady-state assumption is a poor approximation to the equations of motion, but this hypothesis deserves close attention as a test of our understanding of the basic physics of the ocean circulation. In this paper we introduce a physically-based statistical inversion approach to estimate the

Exact MAP states and expectations from perfect sampling: Greig, porteous and seheult revisited

In 1989 Greig, Porteous and Seheult showed that the maximum a posteriori (MAP) state can be exactly calculated for degraded binary images. Their interest was in assessing the performance of algorithms used to find the MAP state, such as simulated annealing. A secondary conclusion was that the MAP state, at least in the restricted setting of two-color images, does not provide a robust reconstruction of the true image. That result has been interpreted by some as indicating that the Ising MRF used by GPS is not a good prior model for such images. We show that such a judgement is premature as the MAP state does not well summarize the information in the posterior distribution in this case. In particular, the deviation of the MAP state from the mean, particularly at larger smoothing parameters, shows that the MAP state is not representative of the bulk of feasible reconstructions. We calculate other summary statistics that interpret and display the information in the posterior by implementing full Bayesian infe...

Reservoir offset models for radiocarbon calibration.

The purpose of a reservoir offset is to enable the application of calibration data (mu (theta ), e.g. Stuiver et al. 1998) developed for one reservoir (primary reservoir) to CRAs from another (secondary reservoir), for example the use of a hemispheric offset for terrestrial samples (Barbetti et al. 1995; McCormac et al. 1998; Sparks et al. 1995; Vogel et al. 1986, 1993). The usual approach has been to define the activity of the secondary reservoir as some form of constant offset (with error) from the primary reservoir (e.g. Higham and Hogg 1985; McFadgen and Manning 1990). In this case, all CRAs from a secondary reservoir are given the same offset. The value of this common offset is not known exactly, but any uncertainty in the measured value of the offset corresponds to uncertainty in the common offset for all CRAs. However, the standard procedure for incorporating offset error into CRAs incorrectly allows a different offset for each CRA. The offset for each CRA is incorrectly allowed to vary by the measurement error reported for the offset value. Technically, the offset is incorrectly treated as varying independently from one CRA to the next, when in fact it is a single parameter for the secondary reservoir in question. In light of this, the calibrated date distributions will be incorrect for CRAs where an offset has been applied and the standard approach to offset error treatment has been used. In many cases, the differences between correct and incorrect calibrated date distributions will be insignificant. However, in some cases significant differences may arise and other approaches to treating the error associated with offsets need to be adopted.

Isotope evidence for agricultural extensification reveals how the world's first cities were fed

This study sheds light on the agricultural economy that underpinned the emergence of the first urban centres in northern Mesopotamia. Using δ13C and δ15N values of crop remains from the sites of Tell Sabi Abyad, Tell Zeidan, Hamoukar, Tell Brak and Tell Leilan (6500–2000 cal bc), we reveal that labour-intensive practices such as manuring/middening and water management formed an integral part of the agricultural strategy from the seventh millennium bc. Increased agricultural production to support growing urban populations was achieved by cultivation of larger areas of land, entailing lower manure/midden inputs per unit area—extensification. Our findings paint a nuanced picture of the role of agricultural production in new forms of political centralization. The shift towards lower-input farming most plausibly developed gradually at a household level, but the increased importance of land-based wealth constituted a key potential source of political power, providing the possibility for greater bureaucratic control and contributing to the wider societal changes that accompanied urbanization.

Spontaneous Magnetization in the Plane

The Arak process is a solvable stochastic process which generates coloured patterns in the plane. Patterns are made up of a variable number of random non-intersecting polygons. We show that the distribution of Arak process states is the Gibbs distribution of its states in thermodynamic equilibrium in the grand canonical ensemble. The sequence of Gibbs distributions forms a new model parameterised by temperature. We prove that there is a phase transition in this model, for some non-zero temperature. We illustrate this conclusion with simulation results. We measure the critical exponents of this off-lattice model and find they are consistent with those of the Ising model in two dimensions.

Prior modeling and posterior sampling in impedance imaging

We examine sample based Bayesian inference from impedance imaging data. We report experiments employing low level pixel based priors with mixed discrete and continuous conductivities. Sampling is carried out using Metropolis- Hasting Markov chain Monte Carlo, employing both large scale, Langevin updates, and state-adaptive local updates. Computing likelihood ratios of conductivity distributions involves solving a second order linear partial differential equation. However our simulation is rendered computationally tractable by an update procedure which employs a linearization of the forward map and thereby avoids solving the PDE for those updates which are rejected.

Genealogies from Time-Stamped Sequence Data

This chapter focuses on on-going research into chronology building tools based on genetic data from DNA sequences. By combining genetic information and radiocarbon data from fossil remains it is possible to recover genealogical structures, population size information, mutation rates and, hence, approximate chronologies for genetic trees. Since this is such a new area of research, this chapter provides consideration of the problems that need tackling, and makes a range of suggestions for modelling aspects of them. Detailed explanations of the proposed models are given and insight into the nature and size of the uncertainties associated with the chronological estimates is obtained. An illustrated case study indicates the type of problem that can already be tackled, and shows that uncertainty derives mainly from the genetic model and not the radiocarbon dates.