Bayesian density estimation using Skew Student-t-Normal mixtures

We present a Bayesian approach for modeling heterogeneous data and estimate multimodal densities using mixtures of Skew Student-t-Normal distributions [Gomez, H.W., Venegas, O., Bolfarine, H., 2007. Skew-symmetric distributions generated by the distribution function of the normal distribution. Environmetrics 18, 395-407]. A stochastic representation that is useful for implementing a MCMC-type algorithm and results about existence of posterior moments are obtained. Marginal likelihood approximations are obtained, in order to compare mixture models with different number of component densities. Data sets concerning the Gross Domestic Product per capita (Human Development Report) and body mass index (National Health and Nutrition Examination Survey), previously studied in the related literature, are analyzed. (c) 2008 Elsevier B.V. All rights reserved.

Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Bayesian L-optimal exact design of experiments for biological kinetic models

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Optimization of neural classifiers based on bayesian decision boundaries and idle neurons pruning

In this article we describe a feature extraction algorithm for pattern classification based on Bayesian Decision Boundaries and Pruning techniques. The proposed method is capable of optimizing MLP neural classifiers by retaining those neurons in the hidden layer that realy contribute to correct classification. Also in this article we proposed a method which defines a plausible number of neurons in the hidden layer based on the stem-and-leaf graphics of training samples. Experimental investigation reveals the efficiency of the proposed method. © 2002 IEEE.

Genetic Correlation for Pre-Weaning and Post-Weaning Traits of Mediterranean Buffaloes from Brazil, Estimated By Bayesian Inference

The aim of this study was to estimate genetic, environmental and phenotypic correlation between birth weight (BW) and weight at 205 days age (W205), BW and weight at 365 days age (W365) and W205-W365, using Bayesian inference. The Brazilian Program for Genetic Improvement of Buffaloes provided the data that included 3,883 observations from Mediterranean breed buffaloes. With the purpose to estimate variance and covariance, bivariate analyses were performed using Gibbs sampler that is included in the MTGSAM software. The model for BW, W205 and W365 included additive direct and maternal genetic random effects, maternal environmental random effect and contemporary group as fixed effect. The convergence diagnosis was achieved using Geweke, a method that uses an algorithm implemented in R software through the package Bayesian Output Analysis. The calculated direct genetic correlations were 0.34 (BW-W205), 0.25 (BW-W365) and 0.74 (W205-W365). The environmental correlations were 0.12, 0.11 and 0.72 between BW-W205, BW-W365 and W205-W365, respectively. The phenotypic correlations were low for BW-W205 (0.01) and BW-W365 (0.04), differently than the obtained for W205-W365 with a value of 0.67. The results indicate that BW trait have low genetic, environmental and phenotypic association with the two others traits. The genetic correlation between W205 and W365 was high and suggests that the selection for weight at around 205 days could be beneficial to accelerate the genetic gain.

Genetic Parameters for Growth Traits of Mediterranean Buffaloes from Brazil, Estimated By Bayesian Inference

Quantitative analysis of growth genetic parameters is not available for many breeds of buffaloes making selection and breeding decisions an empirical process that lacks robustness. The objective of this study was to estimate heritability for birth weight (BW), weight at 205 days (W205) and 365 days (W365) of age using Bayesian inference. The Brazilian Program for Genetic Improvement of Buffaloes provided the data. For the traits BW, W205 and W365 of Brazilian Mediterranean buffaloes 5169, 3792 and 3883 observations have been employed for the analysis, respectively. In order to obtain the estimates of variance, univariate analyses were conducted using the Gibbs sampler included in the MTGSAM software. The model for BW, W205 and W365 included additive direct and maternal genetic random effects, random maternal permanent environmental effect and contemporary group that was treated as a fixed effect. The convergence diagnosis was performed employing Geweke, a method that uses an algorithm from the Bayesian Output Analysis package that was implemented using R software environment. The average values for weight traits were 37.6 +/- 4.7 kg for BW, 192.7 +/- 40.3 kg for W205 and 298.6 +/- 67.4 kg for W365. The heritability posterior distributions for direct and maternal effects were symmetric and close to those expected in a normal distribution. Direct heritability estimates obtained using the modes were 0.30 (BW), 0.52 (W205) and 0.54 (W365). The maternal heritability coefficient estimates were 0.31, 0.19 and 0.21 for BW, W205 and W365, respectively. Our data suggests that all growth traits and mainly W205 and W365, have clear potential for yield improvement through direct genetic selection.

Genetic Parameters for Milk Yield and Lactation Length of Crossbred Buffaloes from Brazil by Bayesian Inference

The objective of the study was to estimate heritability and repeatability for milk yield (MY) and lactation length (LL) in buffaloes using Bayesian inference. The Brazilian genetic improvement program of buffalo provided the data that included 628 females, from four herds, born between 1980 and 2003. In order to obtain the estimates of variance, univariate analyses were performed with the Gibbs sampler, using the MTGSAM software. The model for MY and LL included direct genetic additive and permanent environment as random effects, and contemporary groups, milking frequency and calving number as fixed effects. The convergence diagnosis was performed with the Geweke method using an algorithm implemented in R software through the package Bayesian Output Analysis. Average for milk yield and lactation length was 1,546.1 +/- 483.8 kg and 252.3 +/- 42.5 days, respectively. The heritability coefficients were 0.31 (mode), 0.35 (mean) and 0.34 (median) for MY and 0.11 (mode), 0.10 (mean) and 0.10 (median) for LL. The repeatability coefficient (mode) were 0.50 and 0.15 for MY and LL, respectively. Milk yield is the only trait with clear potential for genetic improvement by direct genetic selection. The repeatability for MY indicates that selection based on the first lactation could contribute for an improvement in this trait.

Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework, prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353-365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R. B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362-1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis-Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271-275]. Our algorithm has only one Metropolis-Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146-178; R. J. Patz and B. W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342-366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599-607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.

Essays on Bayesian Inference for Social Networks

This thesis presents Bayesian solutions to inference problems for three types of social network data structures: a single observation of a social network, repeated observations on the same social network, and repeated observations on a social network developing through time. A social network is conceived as being a structure consisting of actors and their social interaction with each other. A common conceptualisation of social networks is to let the actors be represented by nodes in a graph with edges between pairs of nodes that are relationally tied to each other according to some definition. Statistical analysis of social networks is to a large extent concerned with modelling of these relational ties, which lends itself to empirical evaluation. The first paper deals with a family of statistical models for social networks called exponential random graphs that takes various structural features of the network into account. In general, the likelihood functions of exponential random graphs are only known up to a constant of proportionality. A procedure for performing Bayesian inference using Markov chain Monte Carlo (MCMC) methods is presented. The algorithm consists of two basic steps, one in which an ordinary Metropolis-Hastings up-dating step is used, and another in which an importance sampling scheme is used to calculate the acceptance probability of the Metropolis-Hastings step. In paper number two a method for modelling reports given by actors (or other informants) on their social interaction with others is investigated in a Bayesian framework. The model contains two basic ingredients: the unknown network structure and functions that link this unknown network structure to the reports given by the actors. These functions take the form of probit link functions. An intrinsic problem is that the model is not identified, meaning that there are combinations of values on the unknown structure and the parameters in the probit link functions that are observationally equivalent. Instead of using restrictions for achieving identification, it is proposed that the different observationally equivalent combinations of parameters and unknown structure be investigated a posteriori. Estimation of parameters is carried out using Gibbs sampling with a switching devise that enables transitions between posterior modal regions. The main goal of the procedures is to provide tools for comparisons of different model specifications. Papers 3 and 4, propose Bayesian methods for longitudinal social networks. The premise of the models investigated is that overall change in social networks occurs as a consequence of sequences of incremental changes. Models for the evolution of social networks using continuos-time Markov chains are meant to capture these dynamics. Paper 3 presents an MCMC algorithm for exploring the posteriors of parameters for such Markov chains. More specifically, the unobserved evolution of the network in-between observations is explicitly modelled thereby avoiding the need to deal with explicit formulas for the transition probabilities. This enables likelihood based parameter inference in a wider class of network evolution models than has been available before. Paper 4 builds on the proposed inference procedure of Paper 3 and demonstrates how to perform model selection for a class of network evolution models.

A new snowfall detection algorithm for high latitude regions based on a combination of active and passive sensors

Precipitation retrieval over high latitudes, particularly snowfall retrieval over ice and snow, using satellite-based passive microwave spectrometers, is currently an unsolved problem. The challenge results from the large variability of microwave emissivity spectra for snow and ice surfaces, which can mimic, to some degree, the spectral characteristics of snowfall. This work focuses on the investigation of a new snowfall detection algorithm specific for high latitude regions, based on a combination of active and passive sensors able to discriminate between snowing and non snowing areas. The space-borne Cloud Profiling Radar (on CloudSat), the Advanced Microwave Sensor units A and B (on NOAA-16) and the infrared spectrometer MODIS (on AQUA) have been co-located for 365 days, from October 1st 2006 to September 30th, 2007. CloudSat products have been used as truth to calibrate and validate all the proposed algorithms. The methodological approach followed can be summarised into two different steps. In a first step, an empirical search for a threshold, aimed at discriminating the case of no snow, was performed, following Kongoli et al. [2003]. This single-channel approach has not produced appropriate results, a more statistically sound approach was attempted. Two different techniques, which allow to compute the probability above and below a Brightness Temperature (BT) threshold, have been used on the available data. The first technique is based upon a Logistic Distribution to represent the probability of Snow given the predictors. The second technique, defined Bayesian Multivariate Binary Predictor (BMBP), is a fully Bayesian technique not requiring any hypothesis on the shape of the probabilistic model (such as for instance the Logistic), which only requires the estimation of the BT thresholds. The results obtained show that both methods proposed are able to discriminate snowing and non snowing condition over the Polar regions with a probability of correct detection larger than 0.5, highlighting the importance of a multispectral approach.

A Fully Bayesian Approach for Combining Multilevel Failure Information in Fault Tree Quantification and Corresponding Optimal Resource Allocation

This paper presents a fully Bayesian approach that simultaneously combines basic event and statistically independent higher event-level failure data in fault tree quantification. Such higher-level data could correspond to train, sub-system or system failure events. The full Bayesian approach also allows the highest-level data that are usually available for existing facilities to be automatically propagated to lower levels. A simple example illustrates the proposed approach. The optimal allocation of resources for collecting additional data from a choice of different level events is also presented. The optimization is achieved using a genetic algorithm.

Bayesian Hidden Markov Modeling of Array CGH Data

Genomic alterations have been linked to the development and progression of cancer. The technique of Comparative Genomic Hybridization (CGH) yields data consisting of fluorescence intensity ratios of test and reference DNA samples. The intensity ratios provide information about the number of copies in DNA. Practical issues such as the contamination of tumor cells in tissue specimens and normalization errors necessitate the use of statistics for learning about the genomic alterations from array-CGH data. As increasing amounts of array CGH data become available, there is a growing need for automated algorithms for characterizing genomic profiles. Specifically, there is a need for algorithms that can identify gains and losses in the number of copies based on statistical considerations, rather than merely detect trends in the data. We adopt a Bayesian approach, relying on the hidden Markov model to account for the inherent dependence in the intensity ratios. Posterior inferences are made about gains and losses in copy number. Localized amplifications (associated with oncogene mutations) and deletions (associated with mutations of tumor suppressors) are identified using posterior probabilities. Global trends such as extended regions of altered copy number are detected. Since the posterior distribution is analytically intractable, we implement a Metropolis-within-Gibbs algorithm for efficient simulation-based inference. Publicly available data on pancreatic adenocarcinoma, glioblastoma multiforme and breast cancer are analyzed, and comparisons are made with some widely-used algorithms to illustrate the reliability and success of the technique.

Bayesian change-point analysis in linear regression model with scale mixtures of normal distributions

In this thesis, we consider Bayesian inference on the detection of variance change-point models with scale mixtures of normal (for short SMN) distributions. This class of distributions is symmetric and thick-tailed and includes as special cases: Gaussian, Student-t, contaminated normal, and slash distributions. The proposed models provide greater flexibility to analyze a lot of practical data, which often show heavy-tail and may not satisfy the normal assumption. As to the Bayesian analysis, we specify some prior distributions for the unknown parameters in the variance change-point models with the SMN distributions. Due to the complexity of the joint posterior distribution, we propose an efficient Gibbs-type with Metropolis- Hastings sampling algorithm for posterior Bayesian inference. Thereafter, following the idea of [1], we consider the problems of the single and multiple change-point detections. The performance of the proposed procedures is illustrated and analyzed by simulation studies. A real application to the closing price data of U.S. stock market has been analyzed for illustrative purposes.

Polytomies and bayesian phylogenetic inference

Bayesian phylogenetic analyses are now very popular in systematics and molecular evolution because they allow the use of much more realistic models than currently possible with maximum likelihood methods. There are, however, a growing number of examples in which large Bayesian posterior clade probabilities are associated with very short edge lengths and low values for non-Bayesian measures of support such as nonparametric bootstrapping. For the four-taxon case when the true tree is the star phylogeny, Bayesian analyses become increasingly unpredictable in their preference for one of the three possible resolved tree topologies as data set size increases. This leads to the prediction that hard (or near-hard) polytomies in nature will cause unpredictable behavior in Bayesian analyses, with arbitrary resolutions of the polytomy receiving very high posterior probabilities in some cases. We present a simple solution to this problem involving a reversible-jump Markov chain Monte Carlo (MCMC) algorithm that allows exploration of all of tree space, including unresolved tree topologies with one or more polytomies. The reversible-jump MCMC approach allows prior distributions to place some weight on less-resolved tree topologies, which eliminates misleadingly high posteriors associated with arbitrary resolutions of hard polytomies. Fortunately, assigning some prior probability to polytomous tree topologies does not appear to come with a significant cost in terms of the ability to assess the level of support for edges that do exist in the true tree. Methods are discussed for applying arbitrary prior distributions to tree topologies of varying resolution, and an empirical example showing evidence of polytomies is analyzed and discussed.

Polytomies and bayesian phylogenetic inference

Bayesian phylogenetic analyses are now very popular in systematics and molecular evolution because they allow the use of much more realistic models than currently possible with maximum likelihood methods. There are, however, a growing number of examples in which large Bayesian posterior clade probabilities are associated with very short edge lengths and low values for non-Bayesian measures of support such as nonparametric bootstrapping. For the four-taxon case when the true tree is the star phylogeny, Bayesian analyses become increasingly unpredictable in their preference for one of the three possible resolved tree topologies as data set size increases. This leads to the prediction that hard (or near-hard) polytomies in nature will cause unpredictable behavior in Bayesian analyses, with arbitrary resolutions of the polytomy receiving very high posterior probabilities in some cases. We present a simple solution to this problem involving a reversible-jump Markov chain Monte Carlo (MCMC) algorithm that allows exploration of all of tree space, including unresolved tree topologies with one or more polytomies. The reversible-jump MCMC approach allows prior distributions to place some weight on less-resolved tree topologies, which eliminates misleadingly high posteriors associated with arbitrary resolutions of hard polytomies. Fortunately, assigning some prior probability to polytomous tree topologies does not appear to come with a significant cost in terms of the ability to assess the level of support for edges that do exist in the true tree. Methods are discussed for applying arbitrary prior distributions to tree topologies of varying resolution, and an empirical example showing evidence of polytomies is analyzed and discussed.