This paper proposes approaches for the analysis of multiple changepoint models when dependency in the data is modelled through a hierarchical Gaussian Markov random field. Integrated nested Laplace approximations are used to approximate data quantities, and an approximate filtering recursions approach is proposed for savings in compuational cost when detecting changepoints. All of these methods are simulation free. Analysis of real data demonstrates the usefulness of the approach in general. The new models which allow for data dependence are compared with conventional models where data within segments is assumed independent.

Whether the number of tropical cyclones (TCs) has increased in the last 150 years has become a matter of intense debate. We investigate the effects of beliefs about TC detection capacities in the North Atlantic on trends in TC numbers since the 1870s. While raw data show an increasing trend of TC counts, the capability to detect TCs and to determine intensities and changes in intensity has also increased dramatically over the same period. We present a model of TC activity that allows investigating the relationship between what one believes about the increase in detection and what one believes about TC trends. Previous work has used assumptions on TC tracks, detection capacities or the relationship between TC activity and various climate parameters to provide estimates of year-by-year missed TCs. These estimates and the associated conclusions about trends cover a wide range of possibilities. We build on previous work to investigate the sensitivity of these conclusions to the assumed priors about detection. Our analysis shows that any inference on TC count trends is strongly sensitive to one's specification of prior beliefs about TC detection. Overall, we regard the evidence on the trend in North Atlantic TC numbers to be ambiguous.

The technological progress of the last decades has made a huge amount of information available, often expressed in unconventional formats. Among these, three-way data occur in different application domains from the simultaneous observation of various attributes on a set of units in different situations or locations. These include data coming from longitudinal studies of multiple responses, spatio-temporal data or data collecting multivariate repeated measures. In this work we propose model based clustering for the wide class of continuous three-way data by a general mixture model which can be adapted to the different kinds of three-way data. In so doing we also provide a tool for simultaneously performing model estimation and model selection. The effectiveness of the proposed method is illustrated on a simulation study and on real examples.

A novel approach to evidence assessment in Bayesian hypothesis testing is proposed, in the form of a "neutral-data comparison." The proposed assessment is similar to a Bayes factor, but, rather than comparing posterior to prior odds, it compares the posterior odds of the observed data to that calculated on "neutral" data, which arise as part of the elicitation of prior knowledge. The article develops a general theory of neutral-data comparisons, motivated largely by the Jeffreys-Lindley paradox, and develops methodology for specifying and working with neutral data in the context of Gaussian linear-models analysis. The proposed methodology is shown to exhibit exceptionally strong asymptotic-consistency properties in high dimensions, and, in an application example, to accommodate challenging analysis objectives using basic computational algorithms.

We discuss the development and application of dynamic graphical models for multivariate financial time series in the context of Financial Index Models. The use of graphs generalizes the independence residual variation assumption of index models with a more complex yet still parsimonious alternative. Working with the dynamic matrix-variate graphical model framework, we develop general time-varying index models that are analytically tractable. In terms of methodology, we carefully explore strategies to deal with graph uncertainty and discuss the implementation of a novel computational tool to sequentially learn about the conditional independence relationships defining the model. Additionally, motivated by our applied context, we extend the DGM framework to accommodate random regressors. Finally, in a case study involving 100 stocks, we show that our proposed methodology is able to generate improvements in covariance forecasting and portfolio optimization problems.

We introduce a Bayesian inference mechanism for outlier detection using the augmented Dirichlet process mixture. Outliers are detected by forming a maximum a posteriori (MAP) estimate of the data partition. Observations that comprise small or singleton clusters in the estimated partition are considered outliers. We offer a novel interpretation of the Dirichlet process precision parameter, and demonstrate its utility in outlier detection problems. The precision parameter is used to form an outlier detection criterion based on the Bayes factor for an outlier partition versus a class of partitions with fewer or no outliers. We further introduce a computational method for MAP estimation that is free of posterior sampling, and guaranteed to find a MAP estimate in finite time. The novel methods are compared with several established strategies in a yeast microarray time series.

In sequentially observed data, Bayesian partition models aim at partitioning the entire observation period into disjoint clusters. Each cluster is an aggregation of sequential observations and a simple model is adopted within each cluster. The main inferential problem is the estimation of the number and locations of the clusters. We extend the well-known product partition model (PPM) by assuming that observations within the same cluster have their distributions indexed by correlated and different parameters. Such parameters are similar within a cluster by means of a Gibbs prior distribution. We carried out several simulations and real data set analyses showing that our model provides better estimates for all parameters, including the number and position of the temporal clusters, even for situations favoring the PPM. A free and open source code is available.

Solutions of the bivariate, linear errors-in-variables estimation problem with unspecified errors are expected to be invariant under interchange and scaling of the coordinates. The appealing model of normally distributed true values and errors is unidentified without additional information. I propose a prior density that incorporates the fact that the slope and variance parameters together determine the covariance matrix of the unobserved true values but is otherwise diffuse. The marginal posterior density of the slope is invariant to interchange and scaling of the coordinates and depends on the data only through the sample correlation coefficient and ratio of standard deviations. It covers the interval between the two ordinary least squares estimates but diminishes rapidly outside of it. I introduce the R package leivfor computing the posterior density, and I apply it to examples in astronomy and method comparison.

We consider a statistical model for pairs of traded assets, based on a Cointegrated Vector Auto Regression (CVAR) Model. We extend standard CVAR models to incorporate estimation of model parameters in the presence of price series level shifts which are not accurately modeled in the standard Gaussian error correction model (ECM) framework. This involves developing a novel matrix-variate Bayesian CVAR mixture model, comprised of Gaussian errors intra-day and $\alpha$-stable errors inter-day in the ECM framework. To achieve this we derive conjugate posterior models for the Scale Mixtures of Normals (SMiN CVAR) representation of $\alpha$-stable inter-day innovations. These results are generalized to asymmetric intractable models for the innovation noise at inter-day boundaries allowing for skewed $\alpha$-stable models via Approximate Bayesian computation.

Our proposed model and sampling methodology is general, incorporating the current CVAR literature on Gaussian models, whilst allowing for price series level shifts to occur either at random estimated time points or known \textit{a priori} time points. We focus analysis on regularly observed non-Gaussian level shifts that can have significant effect on estimation performance in statistical models failing to account for such level shifts, such as at the close and open times of markets. We illustrate our model and the corresponding estimation procedures we develop on both synthetic and real data. The real data analysis investigates Australian dollar, Canadian dollar, five and ten year notes (bonds) and NASDAQ price series. In two studies we demonstrate the suitability of statistically modeling the heavy tailed noise processes for inter-day price shifts via an $\alpha$-stable model. Then we fit the novel Bayesian matrix variate CVAR model developed, which incorporates a composite noise model for $\alpha$-stable and matrix variate Gaussian errors, under both symmetric and non-symmetric $\alpha$-stable assumptions.

To determine the size of the drug-involved offender population that could be served effectively and efficiently by partnerships between courts and treatment in the United States, a synthetic dataset is created by Bhati and Roman (2009). Because of hidden structure and aggregation necessary to create variables, there exists latent variance that can not be explained fully by a normal random effect model. Semiparametric regression is a well-known and frequently used tool to capture the functional dependence between variables with fixed effect parametric and nonlinear regression. A new Gibbs sampler is developed here for the number and positions of knots in regression splines by expressing semiparametric regression as a linear mixed model with a random effect term for the basis functions. Our Gibbs sampler exploits the properties of the multinomial-Dirichlet distribution, and is shown to be an improvement, in terms of operator norm and efficiency, over add/delete one MCMC algorithms. We find that the Dirichlet distribution with small values of the parameters results in a smaller number of knots and, in general, good fit to the data. This approach is shown to reveal previously unseen structures in the synthetic dataset of Bhati and Roman.

In this paper we propose a class of prior distributions on decomposable graphs, allowing for improved modeling flexibility. While existing methods solely penalize the number of edges, the proposed work empowers practitioners to control clustering, level of separation, and other features of the graph. Emphasis is placed on a particular prior distribution which derives its motivation from the class of product partition models; the properties of this prior relative to existing priors are examined through theory and simulation. We then demonstrate the use of graphical models in the field of agriculture, showing how the proposed prior distribution alleviates the inflexibility of previous approaches in properly modeling the interactions between the yield of different crop varieties. Lastly, we explore American voting data, comparing the voting patterns amongst the states over the last century.

We develop strategies for mean field variational Bayes approximate inference for Bayesian hierarchical models containing elaborate distributions. We loosely define elaborate distributions to be those having more complicated forms compared with common distributions such as those in the Normal and Gamma families. Examples are Asymmetric Laplace, Skew Normal and Generalized Extreme Value distributions. Such models suffer from the difficulty that the parameter updates do not admit closed form solutions. We circumvent this problem through a combination of (a) specially tailored auxiliary variables, (b) univariate quadrature schemes and (c) finite mixture approximations of troublesome density functions. An accuracy assessment is conducted and the new methodology is illustrated in an application.

This paper presents a method for Bayesian nonparametric analysis of the return distribution in a stochastic volatility model. The distribution of the logarithm of the squared return is flexibly modelled using an infinite mixture of Normal distributions. This allows efficient Markov chain Monte Carlo methods to be developed. Links between the return distribution and the distribution of the logarithm of the squared returns are discussed. The method is applied to simulated data, one asset return series and one stock index return series. We find that estimates of volatility using the model can differ dramatically from those using a Normal return distribution if there is evidence of a heavy-tailed return distribution.