In addition to being crucial to the establishment of archaeological chronologies, radiocarbon dating is vital to the establishment of time lines for many Holocene and late Pleistocene palaeoclimatic studies and palaeoenvironmental reconstructions. The calibration curves necessary to map radiocarbon to calendar ages were originally estimated using only measurements on known age tree-rings. More recently, however, the types of records available for calibration have diversified and a large group of scientists (known as the IntCal Working Group---IWG) with a wide range of backgrounds has come together to create internationally-agreed estimates of the calibration curves. In 2002, Caitlin Buck was recruited to the IWG and asked to offer advice on statistical methods for curve construction. In collaboration with Paul Blackwell, she devised a tailor-made Bayesian curve estimation method which was adopted by the IWG for making all of the 2004 internationally-agreed radiocarbon calibration curve estimates. This paper reports on that work and on the on-going work that will eventually provide models, methods and software for rolling updates to the curve estimates.

This paper presents several new results on Bayesian sample size determination for estimating binomial proportions, and provides a comprehensive comparative overview of the subject. We investigate the binomial sample size problem using generalized versions of the Average Length and Average Coverage Criteria, the Median Length and Median Coverage Criteria, as well as the Worst Outcome Criterion and its modified version. We compare sample sizes derived from highest posterior density and equal-tailed credible intervals. In some cases, we derive, for the first time, closed form sample size formulae, and where this is not possible, we describe various numerical approaches. These range in complexity from Monte Carlo simulations to more sophisticated curve fitting techniques, third order analytic approximations, and exact, but more computationally-intensive, methods. We compare the accuracy and efficiency of the different computational methods for each of the criteria and make recommendations about which methods are preferred. Finally, we consider, again for the first time, issues surrounding prior robustness on the choice of sample size. Examples are given throughout the text.

We introduce a new class of distributions to model directional data, based on hyperspherical log-splines. The class is very flexible and can be used to model data that exhibit features that cannot be accommodated by typical parametric distributions, such as asymmetries and multimodality. The distributions are defined on hyperspheres of any dimension and thus, include the most common circular and spherical cases. Due to the flexibility of hyperspherical log-splines, the distributions can closely approximate observed behaviour and are as smooth as desired. We propose a Bayesian setup for conducting inference with directional log-spline distributions where we pay particular attention to the prior specification and the matching of the priors of the log-splines model and an alternative model constructed through a mixture of von~Mises distributions. We compare both models in the context of three data sets: simulated data on the circle, circular data on the movement of turtles and a spherical application to the arrival direction of cosmic rays.

We analyze complete sequences of successes (hits, walks, and sacrifices) for a group of players from the American and National Leagues, collected over 4 seasons. The goal is to describe how players' performances vary from season to season. In particular, we wish to assess and compare the effect of available occasion-specific covariates over seasons. The data are binary sequences for each player and each season. We model dependence in the binary sequence by an autoregressive logistic model. The model includes lagged terms up to a fixed order. For each player and season we introduce a different set of autologistic regression coefficients, i.e., the regression coefficients are random effects that are specific to each season and player. We use a nonparametric approach to define a random effects distribution. The nonparametric model is defined as a mixture with a Dirichlet process prior for the mixing measure. The described model is justified by a representation theorem for order-$k$ exchangeable sequences. Besides the repeated measurements for each season and player, multiple seasons within a given player define an additional level of repeated measurements. We introduce dependence at this level of repeated measurements by relating the season-specific random effects vectors in an autoregressive fashion. We ultimately conclude that while some covariates like the ERA of the opposing pitcher are always relevant, others like an indicator for the game being into the seventh inning may be significant only for certain seasons, and some others, like the score of the game, can safely be ignored.

Empirical distributions in finance and economics might show heavy tails, volatility clustering, varying mean returns and multimodality as part of their features. However, most statistical models available in the literature assume some kind of parametric form (clearly neglecting important characteristics of the data) or focus on modeling extreme events (therefore, providing no information about the rest of the distribution). In this paper we develop a Bayesian nonparametric prior for a collection of distributions evolving in discrete time. The prior is constructed by defining the distribution at any time point as a Dirichlet process mixture of Gaussian distributions, and inducing dependence through the atoms of their stick-breaking decomposition. A general construction, which allows for trends, periodicities and regressors is described. The resulting model is applied to the estimation of the time-varying travel expense distribution of employees from a major development bank comparable to the IDB, IMF and World Bank.

Based on the fact that any heavy tailed distribution can be approximated by a possibly infinite mixture of Pareto distributions, this paper proposes two Bayesian methodologies tailored to infer on distribution tails belonging to the Frèchet domain of attraction. Firstly, a Bayesian Pareto based clustering procedure is developed, where the mixing distribution is chosen to be the classical conjugate prior of the Pareto distribution. This allows the grouping of $n$ objects into a certain number of clusters according to their extremal behavior and also exhibits a new estimator for the tail index. Secondly, a nonparametric extension of the model based clustering is proposed in which the parameter of interest is the mixing distribution. Estimation of the tail probability is conducted using a Dirichlet process prior for the unknown mixing distribution. To illustrate, both methodologies are applied to simulated data sets and a real data set concerning dietary exposure to a mycotoxin called Ochratoxin A.

This paper proposes a novel approach for modeling prepayment rates of individual pools of mortgages. The model incorporates the empirical evidence that prepayment is past dependent via Bayesian methodology. There are many factors that influence the prepayment behavior and for many of them there is no available (or impossible to gather) information. We implement this issue by creating a Bayesian mixture model and construct a Markov Chain Monte Carlo algorithm to estimate the parameters. We assess the model on a data set from the Bloomberg Database. Our results show that the burnout effect is a significant variable for explaining normal prepayment activities. This result does not hold when prepayment is triggered by non-pool dependent events. We show how to use the new model to compute prices for Mortgage Backed Securities. Monte Carlo simulation is the traditional method for obtaining such prices and the proposed model can be easily incorporated within simulation pricing framework. Prices for standard Pass-Throughs are obtained using simulation.

In Scott (2002) and Congdon (2006), a new method is advanced to compute posterior probabilities of models under consideration. It is based solely on MCMC outputs restricted to single models, i.e., it is bypassing reversible jump and other model exploration techniques. While it is indeed possible to approximate posterior probabilities based solely on MCMC outputs from single models, as demonstrated by Gelfand and Dey (1994) and Bartolucci et al. (2006), we show that the proposals of Scott (2002) and Congdon (2006) are biased and advance several arguments towards this thesis, the primary one being the confusion between model-based posteriors and joint pseudo-posteriors. From a practical point of view, the bias in Scott's (2002) approximation appears to be much more severe than the one in Congdon's (2006), the latter being often of the same magnitude as the posterior probability it approximates, although we also exhibit an example where the divergence from the true posterior probability is extreme.