Showing papers in "Bayesian Analysis in 2011"
TL;DR: Christen et al. as discussed by the authors used a gamma-to-regressive semiparametric model with an arbitrary number of subdivisions along the sediment to estimate the age of sediment cores.
Abstract: Radiocarbon dating is routinely used in paleoecology to build chronolo- gies of lake and peat sediments, aiming at inferring a model that would relate the sediment depth with its age. We present a new approach for chronology building (called \Bacon") that has received enthusiastic attention by paleoecologists. Our methodology is based on controlling core accumulation rates using a gamma au- toregressive semiparametric model with an arbitrary number of subdivisions along the sediment. Using prior knowledge about accumulation rates is crucial and in- formative priors are routinely used. Since many sediment cores are currently ana- lyzed, using difierent data sets and prior distributions, a robust (adaptive) MCMC is very useful. We use the t-walk (Christen and Fox, 2010), a self adjusting, robust MCMC sampling algorithm, that works acceptably well in many situations. Out- liers are also addressed using a recent approach that considers a Student-t model for radiocarbon data. Two examples are presented here, that of a peat core and a core from a lake, and our results are compared with other approaches. Past climates and environments can be reconstructed from deposits such as ocean or lake sediments, ice sheets and peat bogs. Within a vertical sediment proflle (core), mea- surements of microfossils, macrofossils, isotopes and other variables at a range of depths serve as proxy estimates or \proxies" of climate and environmental conditions when the sediment of those depths was deposited. It is crucial to establish reliable relationships between these depths and their ages. Age-depth relationships are used to study the evolution of climate/environmental proxies along sediment depth and therefore through time (e.g., Lowe and Walker 1997). Age-depth models are constructed in various ways. For sediment depths containing organic matter, and for ages younger than c. 50,000 years, radiocarbon dating is often used to create an age-depth model. Cores are divided into slices and some of these are radiocarbon dated. A curve is fltted to the radiocarbon data and interpolated to obtain an age estimate for every depth of the core. The flrst restriction to be considered is that age should be increasing monotonically with depth, because sediment can never have accumulated backwards in time (extraordinary events leading to mixed or reversed sediments are, most of the time, noticeable in the stratigraphy and therefore such cores are ruled out from further analyses). Moreover, cores may have missing sections, leading to ∞at parts in the age depth models.
2,591 citations
TL;DR: A novel class of Bayesian nonparametric priors based on stick-breaking constructions where the weights of the process are constructed as probit transformations of normal random variables, allowing a great variety of models while preserving computational simplicity.
Abstract: We describe a novel class of Bayesian nonparametric priors based on stick-breaking constructions where the weights of the process are constructed as probit transformations of normal random variables. We show that these priors are extremely flexible, allowing us to generate a great variety of models while preserving computational simplicity. Particular emphasis is placed on the construction of rich temporal and spatial processes, which are applied to two problems in finance and ecology.
220 citations
TL;DR: A latent variable representation of regularized support vector machines that enables EM, ECME or MCMC algorithms to provide parameter estimates and shows how to implementing SVM’s with spike-and-slab priors and running them against data from a standard spam filtering data set.
Abstract: Summary This paper presents a latent variable representation of regularized support vector machines (SVM’s) that enables EM, ECME or MCMC algorithms to provide parameter estimates. We verify our representation by demonstrating that minimizing the SVM optimality criterion together with the parameter regularization penalty is equivalent to finding the mode of a mean-variance mixture of normals pseudo-posterior distribution. The latent variables in the mixture representation lead to EM and ECME point estimates of SVM parameters, as well as MCMC algorithms based on Gibbs sampling that can bring Bayesian tools for Gaussian linear models to bear on SVM’s. We show how to implementing SVM’s with spike-and-slab priors and running them against data from a standard spam filtering data set.
199 citations
TL;DR: In this article, the authors develop strategies for mean-field variational Bayes approximate inference for Bayesian hierarchical models containing elaborate distributions, such as Asymmetric Laplace, Skew Normal and Generalized Ex-tree Value distributions.
Abstract: We develop strategies for mean eld variational Bayes approximate inference for Bayesian hierarchical models containing elaborate distributions. We loosely dene 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 Ex- treme Value distributions. Such models suer from the diculty that the param- eter 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) nite mixture approximations of troublesome den-
148 citations
TL;DR: Novel likelihood-free approaches to model comparison are presented, based upon the independent estimation of the evidence of each model under study, which allow the exploitation of MCMC or SMC algorithms for exploring the parameter space, and that they do not require a sampler able to mix between models.
Abstract: Statistical methods of inference typically require the likelihood function to be computable in a reasonable amount of time. The class of "likelihood-free" methods termed Approximate Bayesian Computation (ABC) is able to eliminate this requirement, replacing the evaluation of the likelihood with simulation from it. Likelihood-free methods have gained in efficiency and popularity in the past few years, following their integration with Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) in order to better explore the parameter space. They have been applied primarily to estimating the parameters of a given model, but can also be used to compare models. Here we present novel likelihood-free approaches to model comparison, based upon the independent estimation of the evidence of each model under study. Key advantages of these approaches over previous techniques are that they allow the exploitation of MCMC or SMC algorithms for exploring the parameter space, and that they do not require a sampler able to mix between models. We validate the proposed methods using a simple exponential family problem before providing a realistic problem from human population genetics: the comparison of different demographic models based upon genetic data from the Y chromosome.
145 citations
TL;DR: An extension of the matrix normal model to accommodate multidimensional data arrays, or tensors is described and a class of array normal distributions is generated by applying a group of multilinear transformations to an array of independent standard normal random variables.
Abstract: Modern datasets are often in the form of matrices or arrays, potentially having correlations along each set of data indices. For example, data involving repeated measurements of several variables over time may exhibit temporal correlation as well as correlation among the variables. A possible model for matrix-valued data is the class of matrix normal distributions, which is parametrized by two covariance matrices, one for each index set of the data. In this article we discuss an extension of the matrix normal model to accommodate multidimensional data arrays, or tensors. We show how a particular array-matrix product can be used to generate the class of array normal distributions having separable covariance structure. We derive some properties of these covariance structures and the corresponding array normal distributions, and show how the array-matrix product can be used to define a semi-conjugate prior distribution and calculate the corresponding posterior distribution. We illustrate the methodology in an analysis of multivariate longitudinal network data which take the form of a four-way array.
143 citations
TL;DR: In this article, a general sensitivity measure based on the Hellinger distance was developed to assess sensitivity of the posterior distributions with respect to changes in the prior distributions for the precision parameters.
Abstract: Generalized linear mixed models (GLMMs) enjoy increasing popularity because of their ability to model correlated observations. Integrated nested Laplace approximations (INLAs) provide a fast implementation of the Bayesian approach to GLMMs. However, sensitivity to prior assumptions on the random effects precision parameters is a potential problem. To quantify the sensitivity to prior assumptions, we develop a general sensitivity measure based on the Hellinger distance to assess sensitivity of the posterior distributions with respect to changes in the prior distributions for the precision parameters. In addition, for model selection we suggest several cross-validatory techniques for Bayesian GLMMs with a dichotomous outcome. Although the proposed methodology holds in greater generality, we make use of the developed methods in the particular context of the well-known salamander mating data. We arrive at various new findings with respect to the best fitting model and the sensitivity of the estimates of the model components.
137 citations
TL;DR: In this paper, an extension of the classical Zellner's $g$-prior to generalized linear models was developed, where any continuous proper hyperprior $f(g)$ can be used, giving rise to a large class of hyper-$g$ priors.
Abstract: We develop an extension of the classical Zellner's $g$-prior to generalized linear models. Any continuous proper hyperprior $f(g)$ can be used, giving rise to a large class of hyper-$g$ priors. Connections with the literature are described in detail. A fast and accurate integrated Laplace approximation of the marginal likelihood makes inference in large model spaces feasible. For posterior parameter estimation we propose an efficient and tuning-free Metropolis-Hastings sampler. The methodology is illustrated with variable selection and automatic covariate transformation in the Pima Indians diabetes data set.
104 citations
TL;DR: It is shown that increasing the number of auxiliary variables dramatically increases the acceptance rates in the MCMC algorithm (compared to basic data augmentation schemes), allowing for rapid convergence and mixing.
Abstract: In this article we examine two relatively new MCMC methods which allow for Bayesian inference in diffusion models. First, the Monte Carlo within Metropolis (MCWM) algorithm (O’Neil, Balding, Becker, Serola and Mollison, 2000) uses an importance sampling approximation for the likelihood and yields a limiting stationary distribution that can be made arbitrarily “close” to the posterior distribution (MCWM is not a standard Metropolis-Hastings algorithm, however). The second method, described in Beaumont (2003) and generalized in Andrieu and Roberts (2009), introduces auxiliary variables and utilizes a standard Metropolis-Hastings algorithm on the enlarged space; this method preserves the original posterior distribution. When applied to diffusion models, this approach can be viewed as a generalization of the popular data augmentation schemes that sample jointly from the missing paths and the parameters of the diffusion volatility. We show that increasing the number of auxiliary variables dramatically increases the acceptance rates in the MCMC algorithm (compared to basic data augmentation schemes), allowing for rapid convergence and mixing. The efficacy of ourapproach is demonstrated in a simulation study of the Cox-Ingersoll-Ross (CIR) and Heston models, and is applied to two well known datasets.
53 citations
TL;DR: It is demonstrated that by defining a hierarchical population based nonparametric prior on the cluster locations scaled by the inverse covariance matrices of the likelihood the authors arrive at a 'sparsity prior' representation which admits a conditionally conjugate prior.
Abstract: We propose a hierarchical Bayesian nonparametric mixture model for clustering when some of the covariates are assumed to be of varying relevance to the clustering problem. This can be thought of as an issue in variable selection for unsupervised learning. We demonstrate that by defining a hierarchical population based nonparametric prior on the cluster locations scaled by the inverse covariance matrices of the likelihood we arrive at a 'sparsity prior' representation which admits a conditionally conjugate prior. This allows us to perform full Gibbs sampling to obtain posterior distributions over parameters of interest including an explicit measure of each covariate's relevance and a distribution over the number of potential clusters present in the data. This also allows for individual cluster specific variable selection. We demonstrate improved inference on a number of canonical problems.
46 citations
TL;DR: In this article, an enriched conjugate prior is proposed to model uncertainty on the marginal and conditionals of the Dirichlet process. But it does not address an analogous lack of exibility of standard conju- gate priors in a parametric setting.
Abstract: The precision parameter plays an important role in the Dirichlet Pro- cess. When assigning a Dirichlet Process prior to the set of probability measures on R k , k > 1, this can be restrictive in the sense that the variability is determined by a single parameter. The aim of this paper is to construct an enrichment of the Dirichlet Process that is more exible with respect to the precision parameter yet still conjugate, starting from the notion of enriched conjugate priors, which have been proposed to address an analogous lack of exibility of standard conju- gate priors in a parametric setting. The resulting enriched conjugate prior allows more exibility in modelling uncertainty on the marginal and conditionals. We describe an enriched urn scheme which characterizes this process and show that it can also be obtained from the stick-breaking representation of the marginal and conditionals. For non atomic base measures, this allows global clustering of the marginal variables and local clustering of the conditional variables. Finally, we consider an application to mixture models that allows for uncertainty between homoskedasticity and heteroskedasticity.
TL;DR: In this article, the authors proposed a matching prior based on higher-order asymptotics and pseudo-likelihoods, which allows one to perform accurate inference on the parameter of interest $R$ only, even for small sample sizes.
Abstract: We address the statistical problem of evaluating $R = P(X \lt Y)$, where $X$ and $Y$ are two independent random variables. Bayesian parametric inference is based on the marginal posterior density of $R$ and has been widely discussed under various distributional assumptions on $X$ and $Y$. This classical approach requires both elicitation of a prior on the complete parameter and numerical integration in order to derive the marginal distribution of $R$. In this paper, we discuss and apply recent advances in Bayesian inference based on higher-order asymptotics and on pseudo-likelihoods, and related matching priors, which allow one to perform accurate inference on the parameter of interest $R$ only, even for small sample sizes. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. From a theoretical point of view, we show that the used prior is a strong matching prior. From an applied point of view, the accuracy of the proposed methodology is illustrated both by numerical studies and by real-life data concerning clinical studies.
TL;DR: This work proposes 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 and provides a tool for simultaneously performing model estimation and model selection.
Abstract: 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.
TL;DR: In this article, the authors present a method for Bayesian nonparametric analysis of the return distribution in a stochastic volatility model, where the distribution of the logarithm of the squared return is flexibly modelled using an infinite mixture of Normal distributions.
Abstract: 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.
TL;DR: The use of graphs generalizes the independence residual variation assumption of index models with a more complex yet still parsimonious alternative and develops general time-varying index models that are analytically tractable.
Abstract: We discuss the development and application of dynamic graphical mod- els for multivariate flnancial 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 methodol- ogy, we carefully explore strategies to deal with graph uncertainty and discuss the implementation of a novel computational tool to sequentially learn about the con- ditional independence relationships deflning 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 pro- posed methodology is able to generate improvements in covariance forecasting and portfolio optimization problems.
TL;DR: In this paper, a hierarchical Gaussian Markov random field (GMRF) is used for the analysis of multiple changepoint models when dependency in the data is modelled through a hierarchical GMRF and integrated nested Laplace approximations are used to approximate data quantities.
Abstract: This paper proposes approaches for the analysis of multiple changepoint models when dependency in the data is modelled through a hierarchical Gaussian Markov random eld. Integrated nested Laplace approximations are used to approximate data quantities, and an approximate ltering 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.
TL;DR: A Bayesian inference mechanism for outlier detection using the augmented Dirichlet process mixture and a computational method for MAP estimation that is free of posterior sampling, and guaranteed to give a MAP estimate in flnite time are introduced.
Abstract: 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 out- liers. We ofier 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 flnd a MAP estimate in flnite time. The novel meth- ods are compared with several established strategies in a yeast microarray time series.
TL;DR: The goal was to determine the cocktails of pesticide residues to which the French population is simultaneously exposed through its current diet in order to study their possible combined effects on health through toxicological experiments.
Abstract: This work introduces a specific application of Bayesian nonparametric statistics to the food risk analysis framework The goal was to determine the cocktails of pesticide residues to which the French population is simultaneously exposed through its current diet in order to study their possible combined effects on health through toxicological experiments To do this, the joint distribution of exposures to a large number of pesticides, which we called the co-exposure distribution, was assessed from the available consumption data and food contamination analyses We proposed modelling the co-exposure using a Dirichlet process mixture based on a multivariate Gaussian kernel so as to determine groups of individuals with similar co-exposure patterns Posterior distributions and the optimal partition were computed through a Gibbs sampler based on stick-breaking priors The study of the correlation matrix of the sub-population co-exposures will be used to define the cocktails of pesticides to which they are jointly exposed at high doses To reduce the computational burden due to the high data dimensionality, a random-block sampling approach was used In addition, we propose to account for the uncertainty of food contamination through the introduction of an additional level of hierarchy in the model The results of both specifications are described and compared
TL;DR: A new tree-based graphical model for modelling discrete-valued discrete-time multivariate processes which are hypothesised to exhibit symmetries in how some intermediate situations might unfold is proposed and a one-step-ahead prediction algorithm is implemented.
Abstract: A new tree-based graphical model | the dynamic staged tree | is proposed for modelling discrete-valued discrete-time multivariate processes which are hypothesised to exhibit symmetries in how some intermediate situations might unfold. We deflne and implement a one-step-ahead prediction algorithm with the model using multi-process modelling and the power steady model that is robust to short-term variations in the data yet sensitive to underlying system changes. We demonstrate that the whole analysis can be performed in a conjugate way so that the potentially vast model space can be traversed quickly and then results communicated transparently. We also demonstrate how to analyse a general set of causal hypotheses on this model class. Our techniques are illustrated using a simple educational example.
TL;DR: In this article, a method is proposed to select relevant variables among tens of thousands in a probit mixed regression model, considered as part of a larger hierarchical Bayesian model, where latent variables are used to identify subsets of selected variables and the grouping (or blocking) technique of Liu (1994) is combined with a Metropolis-within-Gibbs algorithm (Robert and Casella 2004).
Abstract: In computational biology, gene expression datasets are characterized by very few individual samples compared to a large number of measurements per sample. Thus, it is appealing to merge these datasets in order to increase the number of observations and diversify the data, allowing a more reliable selection of genes relevant to the biological problem. Besides, the increased size of a merged dataset facilitates its re-splitting into training and validation sets. This necessitates the introduction of the dataset as a random effect. In this context, extending a work of Lee et al. (2003), a method is proposed to select relevant variables among tens of thousands in a probit mixed regression model, considered as part of a larger hierarchical Bayesian model. Latent variables are used to identify subsets of selected variables and the grouping (or blocking) technique of Liu (1994) is combined with a Metropolis-within-Gibbs algorithm (Robert and Casella 2004). The method is applied to a merged dataset made of three individual gene expression datasets, in which tens of thousands of measurements are available for each of several hundred human breast cancer samples. Even for this large dataset comprised of around 20000 predictors, the method is shown to be efficient and feasible. As an illustration, it is used to select the most important genes that characterize the estrogen receptor status of patients with breast cancer.
TL;DR: A class of prior distributions on decompos- able graphs, allowing for improved modeling exibility, and the use of graphical models in the eld of agriculture, showing how the proposed prior distribution alleviates the inexibility of previous approaches in properly modeling the interactions between the yield of dierent crop varieties.
Abstract: y Abstract. In this paper we propose a class of prior distributions on decompos- able graphs, allowing for improved modeling exibility. 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 eld of agriculture, showing how the proposed prior distribution alleviates the inexibility of previous approaches in properly modeling the interactions between the yield of dierent crop varieties. Lastly, we explore American voting data, comparing the voting patterns amongst the states over the last century.
TL;DR: In this article, an approach for Bayesian modeling in spherical data sets is described based upon a recent construction called the needlet, which is a particular form of spherical wavelet with many favorable statistical and computational properties.
Abstract: This paper describes an approach for Bayesian modeling in spherical data sets. Our method is based upon a recent construction called the needlet, which is a particular form of spherical wavelet with many favorable statistical and computational properties. We perform shrinkage and selection of needlet coefficients, focusing on two main alternatives: empirical-Bayes thresholding, and Bayesian local shrinkage rules. We study the performance of the proposed methodology both on simulated data and on two real data sets: one involving the cosmic microwave background radiation, and one involving the reconstruction of a global news intensity surface inferred from published Reuters articles in August, 1996. The fully Bayesian approach based on robust, sparse shrinkage priors seems to outperform other alternatives.
TL;DR: In this article, a matrix-variate Bayesian CVAR mixture model is proposed 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.
Abstract: 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.
TL;DR: In this article, the authors investigate the relationship between what one believes about the increase in detection and what one believe about TC trends since the 1870s and show that any inference on TC count trends is strongly sensitive to one's specifcation of prior beliefs about TC detection.
Abstract: Whether the number of tropical cyclones (TCs) has increased in the last 150 years has become a matter of intense debate. We investigate the efiects of beliefs about TC detection capacities in the North Atlantic on trends in TC num- bers 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 activ- ity 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 speciflcation of prior beliefs about TC detection. Overall, we regard the evidence on the trend in North Atlantic TC numbers to be ambiguous.
TL;DR: The well-known product partition model (PPM) is extended by assuming that observations within the same cluster have their distributions in- dexed by correlated and difierent parameters, which are similar within a cluster by means of a Gibbs prior distribution.
Abstract: In sequentially observed data, Bayesian partition models aim at par- titioning 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 loca- tions of the clusters. We extend the well-known product partition model (PPM) by assuming that observations within the same cluster have their distributions in- dexed by correlated and difierent parameters. Such parameters are similar within a cluster by means of a Gibbs prior distribution. We carried out several simula- tions 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.
TL;DR: In this paper, a Bayesian approach to functional data analysis (FDA) is proposed, combining a special Demmler-Reinsch like basis of interpolation splines to represent functions parsimoniously and flexibly, and latent variable models initially introduced for probabilistic principal components analysis or canonical correlation analysis of the corresponding coefficients.
Abstract: In functional data analysis (FDA) it is of interest to generalize techniques of multivariate analysis like canonical correlation analysis or regression to functions which are often observed with noise. In the proposed Bayesian approach to FDA two tools are combined: (i) a special Demmler-Reinsch like basis of interpolation splines to represent functions parsimoniously and flexibly; (ii) latent variable models initially introduced for probabilistic principal components analysis or canonical correlation analysis of the corresponding coefficients. In this way partial curves and non-Gaussian measurement error schemes can be handled. Bayesian inference is based on a variational algorithm such that computations are straight forward and fast corresponding to an idea of FDA as a toolbox for explorative data analysis. The performance of the approach is illustrated with synthetic and real data sets.
TL;DR: In this paper, a Bayesian surrogate model for the analysis of periodic or quasi-periodic time series data is presented and a computationally efficient implementation that enables Bayesian model comparison is described.
Abstract: We present a Bayesian surrogate model for the analysis of periodic or quasi-periodic time series data. We describe a computationally efficient implementation that enables Bayesian model comparison. We apply this model to simulated and real exoplanet observations. We discuss the results and demonstrate some of the challenges for applying our surrogate model to realistic exoplanet data sets. In particular, we find that analyses of real world data should pay careful attention to the effects of uneven spacing of observations and the choice of prior for the "jitter" parameter.
TL;DR: In this paper, the authors proposed 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.
Abstract: 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.
TL;DR: In this article, the authors propose the calibrated posterior predictive $p$-value as an interpretable goodness-of-fit (GOF) measure for regression models in sequential regression multiple imputation (SRMI).
Abstract: We propose the calibrated posterior predictive $p$-value ($cppp$) as an interpretable goodness-of-fit (GOF) measure for regression models in sequential regression multiple imputation (SRMI). The $cppp$ is uniformly distributed under the assumed model, while the posterior predictive $p$-value ($ppp$) is not in general and in particular when the percentage of missing data, $pm$, increases. Uniformity of $cppp$ allows the analyst to evaluate properly the evidence against the assumed model. We show the advantages of $cppp$ over $ppp$ in terms of power in detecting common departures from the assumed model and, more importantly, in terms of robustness with respect to $pm$. In the imputation phase, which provides a complete database for general statistical analyses, default and improper priors are usually needed, whereas the $cppp$ requires a proper prior on regression parameters. We avoid this problem by introducing the use of a minimum training sample that turns the improper prior into a proper distribution. The dependency on the training sample is naturally accounted for by changing the training sample at each step of the SRMI. Our results come from theoretical considerations together with simulation studies and an application to a real data set of anthropometric measures.