Marginal Posterior Distributions Research Articles

Bayesian Inference for Mixed Model-Based Genome-Wide Analysis of Expression Quantitative Trait Loci by Gibbs Sampling.

The importance of expression quantitative trait locus (eQTL) has been emphasized in understanding the genetic basis of cellular activities and complex phenotypes. Mixed models can be employed to effectively identify eQTLs by explaining polygenic effects. In these mixed models, the polygenic effects are considered as random variables, and their variability is explained by the polygenic variance component. The polygenic and residual variance components are first estimated, and then eQTL effects are estimated depending on the variance component estimates within the frequentist mixed model framework. The Bayesian approach to the mixed model-based genome-wide eQTL analysis can also be applied to estimate the parameters that exhibit various benefits. Bayesian inferences on unknown parameters are based on their marginal posterior distributions, and the marginalization of the joint posterior distribution is a challenging task. This problem can be solved by employing a numerical algorithm of integrals called Gibbs sampling as a Markov chain Monte Carlo. This article reviews the mixed model-based Bayesian eQTL analysis by Gibbs sampling. Theoretical and practical issues of Bayesian inference are discussed using a concise description of Bayesian modeling and the corresponding Gibbs sampling. The strengths of Bayesian inference are also discussed. Posterior probability distribution in the Bayesian inference reflects uncertainty in unknown parameters. This factor is useful in the context of eQTL analysis where a sample size is too small to apply the frequentist approach. Bayesian inference based on the posterior that reflects prior knowledge, will be increasingly preferred with the accumulation of eQTL data. Extensive use of the mixed model-based Bayesian eQTL analysis will accelerate understanding of eQTLs exhibiting various regulatory functions.

Priors and Posteriors in Bayesian Timing of Divergence Analyses: The Age of Butterflies Revisited.

The need for robust estimates of times of divergence is essential for downstream analyses, yet assessing this robustness is still rare. We generated a time-calibrated genus-level phylogeny of butterflies (Papilionoidea), including 994 taxa, up to 10 gene fragments and an unprecedented set of 12 fossils and 10 host-plant node calibration points. We compared marginal priors and posterior distributions to assess the relative importance of the former on the latter. This approach revealed a strong influence of the set of priors on the root age but for most calibrated nodes posterior distributions shifted from the marginal prior, indicating significant information in the molecular data set. Using a very conservative approach we estimated an origin of butterflies at 107.6 Ma, approximately equivalent to the latest Early Cretaceous, with a credibility interval ranging from 89.5 Ma (mid Late Cretaceous) to 129.5 Ma (mid Early Cretaceous). In addition, we tested the effects of changing fossil calibration priors, tree prior, different sets of calibrations and different sampling fractions but our estimate remained robust to these alternative assumptions. With 994 genera, this tree provides a comprehensive source of secondary calibrations for studies on butterflies.

Inferring physical parameters in solar prominence threads

Context. High resolution observations have permitted the resolution of solar prominences/filaments into sets of threads/fibrils. However, the values of the physical parameters of these threads and their structuring remain poorly constrained. Aims. We use prominence seismology techniques to analyse transverse oscillations in threads by comparing magnetohydrodynamic (MHD) models and observations. Methods. We applied Bayesian methods to obtain two different types of information. We first inferred the marginal posterior distribution of physical parameters such as the magnetic field strength or length of the thread, when a totally filled tube, partially filled tube, and three damping models are considered as certain; the three damping models are resonant absorption in the Alfvén continuum, resonant absorption in the slow continuum, and Cowling’s diffusion. Then, we compared the relative plausibility between alternative MHD models by computing the Bayes factors. Results. Well-constrained probability density distributions can be obtained for the magnetic field strength, length of the thread, density contrast, and parameters associated with the damping models. In a comparison of the damping models of resonant absorption in the Alfvén continuum, resonant absorption in the slow continuum, and Cowling’s diffusion due to partial ionisation of prominence plasma, the resonant absorption in the Alfvén continuum is the most plausible mechanism to explain the existing observations. Relations between periods of fundamental and first overtone kink modes with values around 1 are better explained by expressions of the period ratio in the long thread approximation, while the rest of the values are more probable in the short thread limit for the period ratio. Conclusions. Our results show that Bayesian analysis offers valuable methods to perform parameter inference and a model comparison in the context of prominence seismology.

Bayesian Modelling on Incidence of Pregnancy among HIV/AIDS Patient Women at Adare Hospital, Hawassa, Ethiopia

HIV/AIDS is the most serious diseases human kind has ever faced and a public problem, particularly, for women of childbearing age. For HIV infected women, the prospects of getting pregnant and having an HIV negative baby could be significantly improved with the increasing of the availability of Antiretroviral Therapy (ART). Even though, ART treatment has shown significant effect of clinical importance to reduce the risk of mother to child transmission of HIV but, HIV infected Women remain poorly understood or they fear to be pregnant and having HIV negative child. To the authors’ knowledge, no study examined incidence of pregnancy among women on ART follow-up in Ethiopia. In response, we conducted a study to explore the incidence and potential predictors of pregnancy. The objective of this study was to investigate the incidence of pregnancy among HIV/AIDS patient women under ART follow-up. A retrospective cohort study was conducted based on secondary data that reviews or visits medical chart of HIV/ADIS patient women aged 15-49 years under ART follow-up from April 2008 to February 2015. Out of 720 total patient women, a sample of size 328 was selected by using simple random sampling technique. Bayesian estimation were used for binary logistic regression model to identify the significant factors of incidence of pregnancy. The Gibbs sampler algorithm was implemented by WinBUGS software to solve approximate properties of the marginal posterior distributions for each parameter in Bayesian estimation. The results of this study revealed 21.3% of women got pregnancy during the follow-up. From Bayesian logistic regression analysis, significant predictors of incidence of pregnancy were: WHO clinical stage, marital status, contraception use, number of child alive before ART follow-up, CD4 cell count, time of Antiretroviral Therapy (ART) follow-up, educational level, spouse's HIV status, occupation and age (at p=0.05). In this study, when age of the women increased, the probability of becoming pregnant was decreased and advanced WHO clinical stage were associated with decreased incidence of pregnancy. Time on ART was a strong predictor of becoming pregnant: longer time on ART was associated with increased probability of becoming pregnant. Educational level of women was positively related with incidence of pregnancy that is, women who had college and above educational level was more likely to become pregnant. When CD4 count increased, incidence of pregnancy also increased and married women had more chance to become pregnant. The predictors identified in this study can be used to care for those HIV/AIDS patient women who want to have baby.

The semi-parametric Bernstein-von Mises theorem for regression models with symmetric errors

In a smooth semi-parametric model, the marginal posterior distribution for a finite dimensional parameter of interest is expected to be asymptotically equivalent to the sampling distribution of any efficient point-estimator. The assertion leads to asymptotic equivalence of credible and confidence sets for the parameter of interest and is known as the semi-parametric Bernstein-von Mises theorem. In recent years, it has received much attention and has been applied in many examples. We consider models in which errors with symmetric densities play a role; more specifically, it is shown that the marginal posterior distributions of regression coefficients in the linear regression and linear mixed effect models satisfy the semi-parametric Bernstein-von Mises assertion. As a consequence, Bayes estimators in these models achieve frequentist inferential optimality, as expressed e.g. through Hajek's convolution and asymptotic minimax theorems. Conditions for the prior on the space of error densities are relatively mild and well-known constructions like the Dirichlet process mixture of normal densities and random series priors constitute valid choices. Particularly, the result provides an efficient estimate of regression coefficients in the linear mixed effect model, for which no other efficient point-estimator was known previously.

Posterior asymptotic normality for an individual coordinate in high-dimensional linear regression

We consider the sparse high-dimensional linear regression model $Y=Xb+\epsilon$ where $b$ is a sparse vector. For the Bayesian approach to this problem, many authors have considered the behavior of the posterior distribution when, in truth, $Y=X\beta+\epsilon$ for some given $\beta$. There have been numerous results about the rate at which the posterior distribution concentrates around $\beta$, but few results about the shape of that posterior distribution. We propose a prior distribution for $b$ such that the marginal posterior distribution of an individual coordinate $b_i$ is asymptotically normal centered around an asymptotically efficient estimator, under the truth. Such a result gives Bayesian credible intervals that match with the confidence intervals obtained from an asymptotically efficient estimator for $b_i$. We also discuss ways of obtaining such asymptotically efficient estimators on individual coordinates. We compare the two-step procedure proposed by Zhang and Zhang (2014) and a one-step modified penalization method.

Bayesian inference for extreme value flood frequency analysis in Bangladesh using Hamiltonian Monte Carlo techniques

In Bangladesh, major floods are frequent due to its unique geographic location. About one-fourth to one-third of the country is inundated by overflowing rivers during the monsoon season almost every year. Calculating the risk level of river discharge is important for making plans to protect the ecosystem and increasing crop and fish production. In recent years, several Bayesian Markov chain Monte Carlo (MCMC) methods have been proposed in extreme value analysis (EVA) for assessing the flood risk in a certain location. The Hamiltonian Monte Carlo (HMC) method was employed to obtain the approximations to the posterior marginal distribution of the Generalized Extreme Value (GEV) model by using annual maximum discharges in two major river basins in Bangladesh. The discharge records of the two largest branches of the Ganges-Brahmaputra-Meghna river system in Bangladesh for the past 42 years were analysed. To estimate flood risk, a return level with 95% confidence intervals (CI) has also been calculated. Results show that, the shape parameter of each station was greater than zero, which shows that heavy-tailed Frechet cases. One station, Bahadurabad, at Brahmaputra river basin estimated 141,387 m3s-1 with a 95% CI range of [112,636, 170,138] for 100-year return level and the 1000-year return level was 195,018 m3s-1 with a 95% CI of [122493, 267544]. The other station, Hardinge Bridge, at Ganges basin estimated 124,134 m3 s-1 with a 95% CI of [108,726, 139,543] for 100-year return level and the 1000-year return level was 170,537 m3s-1 with a 95% CI of [133,784, 207,289]. As Bangladesh is a flood prone country, the approach of Bayesian with HMC in EVA can help policy-makers to plan initiatives that could result in preventing damage to both lives and assets.

What to expect when you're calibrating: Measuring the effect of calibration on the estimation of macroeconomic models

I propose two measures of the impact of calibration on the estimation of macroeconomic models. The first quantifies the amount of information introduced with respect to each estimated parameter as a result of fixing the value of one or more calibrated parameters. The second is a measure of the sensitivity of parameter estimates to perturbations in the calibration values. The purpose of the measures is to show researchers how much and in what way calibration affects their estimation results – by shifting the location and reducing the spread of the marginal posterior distributions of the estimated parameters. Such analysis is often appropriate since macroeconomists do not always agree on whether and how to calibrate structural parameters in macroeconomic models. The methodology is illustrated using the models estimated in Smets and Wouters (2007) and Schmitt-Grohé and Uribe (2012).

Adaptive importance sampling for estimating multi-component chirp signal parameters in colored noise

The problem of estimating the center frequency and the chirp rate of multi-component chirp signals from data corrupted with colored noise often occurs in many signal processing applications. In the Bayesian framework, an algorithm based on adaptive importance sampling (AIS) and modified discrete chirp Fourier transform (MDCFT) was introduced in the present paper in order to calculate the center frequency and the chirp rate of chirp signals from observations corrupted by additive autoregressive (AR) random noise. First, importance functions (IFs) were constructed to approximate the marginal posterior distributions of interested parameters with observed data, and AIS was then employed to evaluate the posterior distributions. MDCFT was adopted to detect the spectral peaks of chirp signals, and the obtained results were applied to construct IFs with multivariate student-t distribution. It was found that the quality of the samples generated from multivariate student-t distribution was gradually improved. Finally, computer simulations validated the results obtained from the proposed algorithm. In comparison with the Markov chain-based method, the proposed method had better SNR adaptability and excellent convergence rate and required fewer samples. Furthermore, the proposed algorithm can be considered as a further expansion of previous works related to AIS method for estimating multi-component chirp signal parameters in white noise.

Genetic variation in serum protein pattern and blood β-hydroxybutyrate and their relationships with udder health traits, protein profile, and cheese-making properties in Holstein cows

The aim of this study was to investigate in Holstein cows the genetic basis of blood serum metabolites [i.e., total protein, albumin, globulin, albumin:globulin ratio (A:G), and blood β-hydroxybutyrate (BHB)], a set of milk phenotypes related to udder health, milk quality technological characteristics, and genetic relationships among them. Samples of milk were collected from 498 Holstein cows belonging to 28 herds. All animal welfare and milk phenotypes were assessed using standard analytical methodology. A set of Bayesian univariate and bivariate animal models was implemented via Gibbs sampling, and statistical inference was based on the marginal posterior distributions of parameters of concern. We observed a small additive genetic influence for serum albumin concentrations, moderate heritability (≥0.20) for total proteins, globulins, and A:G, and high heritability (0.37) for blood BHB. Udder health traits (somatic cell score, milk lactose, and milk pH) showed low or moderate heritabilities (0.15-0.20), whereas variations in milk protein fraction concentrations were confirmed as mostly under genetic control (heritability: 0.21-0.71). The moderate and high heritabilities observed for milk coagulation properties and curd firming modeling parameters provided confirmation that genetic background exerts a strong influence on the cheese-making ability of milk, largely due to genetic polymorphisms in the major milk protein genes. Blood BHB showed strong negative genetic correlations with globulins (-0.619) but positive correlations with serum albumin (0.629) and A:G (0.717), which suggests that alterations in the serum protein pattern and BHB blood levels are likely to be genetically related. Strong relationships were found between albumin and fat percentages (-0.894), between globulin and αS2-CN (-0.610), and, to a lesser extent, between serum protein pattern and milk technological characteristics. Genetic relationships between blood BHB and traits related to udder health and milk quality and technological characteristics were mostly weak. This study provides evidence that there is exploitable additive genetic variation for traits related to animal health and welfare and throws light on the shared genetic basis of these traits and the phenotypes related to the quality and cheese-making ability of milk.

Mapping Soil Cation‐Exchange Capacity using Bayesian Modeling and Proximal Sensors at the Field Scale

Core Ideas A Bayesian inference approach (INLA‐SPDE) was used to map topsoil and subsoil CEC. DEM, gamma‐ray spectrometer and EM induction data were combined to map CEC. Posterior marginal distributions of the model parameters and responses were estimated. Gamma‐ray data performed best in the topsoil, followed by DUALEM421S and elevation. Elevation data performed best in the subsoil, followed by gamma‐ray and DUALEM421S. Cation exchange capacity (CEC) affects soil fertility, acidity, and structural resilience. This is particularly the case in sugarcane growing areas of Australia because the soil there is sandy (>60%), strongly acidic (pH < 5.5), and strongly sodic (exchangeable sodium percentage [ESP] > 15%). Unfortunately, obtaining information on CEC at the field extent is time‐consuming and expensive. Here, we used a digital soil mapping approach to add value to limited (40) topsoil (0–0.3 m) and subsoil (0.6–0.9 m) CEC information. We first collected proximally sensed ancillary data from three sources, including a digital elevation model (DEM), γ‐ray (γ‐ray) spectrometer (RS700) and electromagnetic (EM) induction instruments. We then use a Bayesian inference approach (Integrated Nested Laplace Approximation with Stochastic Partial Differential Equation, INLA‐SPDE) implemented in R software to model the CEC and ancillary data. Accuracy (RMSE), bias (ME), and concordance (Lin's) of models were also generated from the different sources of ancillary data, either in combination or alone. We concluded, overall, that the INLA–SPDE approach could provide estimations of the posterior marginal distributions of the model parameters as well as the model responses as reported by other researchers. We also concluded that using the ancillary data sources in combination was most accurate (e.g., RMSE = 0.72) to predict CEC, least biased (e.g., ME = 0.07) and had the highest concordance (e.g., Lin's = 0.69) in both the topsoil and subsoil than using the ancillary data alone. The best ancillary data, when used alone for mapping CEC in the topsoil, was γ‐ray spectrometry, followed by EM data and elevation. For subsoil CEC, it was elevation, followed by γ‐ray spectrometry and then soil electrical conductivity (ECa) data. The maps of the credibility interval (CI) indicated that better predictions were achieved in the topsoil and indicated where improvements in prediction could be achieved in the subsoil.

Bayesian Modeling of Flood Frequency Analysis in Bangladesh Using Hamiltonian Monte Carlo Techniques

In recent years, several Bayesian Markov chain Monte Carlo (MCMC) methods have been proposed in extreme value analysis (EVA) for assessing the flood risk in a certain location. In this study, the Hamiltonian Monte Carlo (HMC) method was employed to obtain the approximations to the posterior marginal distribution of the Generalized Extreme Value (GEV) model by using annual maximum discharges in two major river basins in Bangladesh. As a comparison, the well-known Metropolis-Hasting (MH) algorithm was also applied, but did not converge well and yielded skewness values opposite those of HMC and the statistical characteristic of the data sets. The discharge records of the Ganges and Brahmaputra rivers in Bangladesh for the past 42 years were analyzed. To estimate flood risk, a return level with 95% confidence intervals (CI) has also been calculated. Results show that the shape parameter of each station was greater than zero, which describes the heavy-tailed Fréchet cases of the GEV distributions. One station, Bahadurabad in the Brahmaputra river basin, estimated 141,387 m3·s−1 with a 95% CI range of [112,636, 170,138] for the 100-year return level, and the 1000-year return level was 195,018 m3·s−1 with a 95% CI of [122,493, 267,544]. The other station, Hardinge Bridge at the Ganges basin, estimated 124,134 m3·s−1 with a 95% CI of [108,726, 139,543] for the 100-year return level, and the 1000-year return level was 170,537 m3·s−1 with a 95% CI of [133,784, 207,289]. As Bangladesh is a flood-prone country, the approach of Bayesian with HMC in EVA can help policy-makers to plan initiatives that could result in preventing damage to both lives and assets.

SAFlex: A structural alphabet extension to integrate protein structural flexibility and missing data information.

In this paper, we describe SAFlex (Structural Alphabet Flexibility), an extension of an existing structural alphabet (HMM-SA), to better explore increasing protein three dimensional structure information by encoding conformations of proteins in case of missing residues or uncertainties. An SA aims to reduce three dimensional conformations of proteins as well as their analysis and comparison complexity by simplifying any conformation in a series of structural letters. Our methodology presents several novelties. Firstly, it can account for the encoding uncertainty by providing a wide range of encoding options: the maximum a posteriori, the marginal posterior distribution, and the effective number of letters at each given position. Secondly, our new algorithm deals with the missing data in the protein structure files (concerning more than 75% of the proteins from the Protein Data Bank) in a rigorous probabilistic framework. Thirdly, SAFlex is able to encode and to build a consensus encoding from different replicates of a single protein such as several homomer chains. This allows localizing structural differences between different chains and detecting structural variability, which is essential for protein flexibility identification. These improvements are illustrated on different proteins, such as the crystal structure of an eukaryotic small heat shock protein. They are promising to explore increasing protein redundancy data and obtain useful quantification of their flexibility.

Estimating Seasonally Frozen Ground Depth From Historical Climate Data and Site Measurements Using a Bayesian Model

AbstractWe develop a Bayesian model to predict the maximum thickness of seasonally frozen ground (MTSFG) using historical air temperature and precipitation observations. We use the Stefan solution and meteorological data from 11 stations to estimate the MTSFG changes from 1961 to 2016 in the Yellow River source region of northwestern China. We employ an antecedent precipitation index model to estimate changes in the liquid soil water content. The marginal posterior probability distributions of the antecedent precipitation index parameters are estimated using Markov chain Monte Carlo sampling methods. We compare the results of our stochastic method with those obtained from the traditional deterministic method and find that they are consistent in general. The stochastic approach is effective for estimating the historical changes in the frozen ground depth (root‐mean‐square errors = 0.13–0.35 m), and it provides more information on model uncertainty regarding soil moisture variations. Additionally, simulation shows that the MTSFG has decreased by 0.31 cm per year over the last 56 years on the northeastern Qinghai‐Tibet Plateau. This decrease in frost depth accelerated in the 1990s and 2000s. Considering the lack of data on seasonally frozen soil monitoring, the Bayesian method provides a pragmatic approach to statistically model frozen ground changes using available meteorological data.

Bayesian estimation of direct and correlated responses to selection on linear or ratio expressions of feed efficiency in pigs

BackgroundThis study aimed at (1) deriving Bayesian methods to predict breeding values for ratio (i.e. feed conversion ratio; FCR) or linear (i.e. residual feed intake; RFI) traits; (2) estimating genetic parameters for average daily feed consumption (ADFI), average daily weight gain (ADG), lean meat percentage (LMP) along with the derived traits of RFI and FCR; and (3) deriving Bayesian estimates of direct and correlated responses to selection on RFI, FCR, ADG, ADFI, and LMP. Response to selection was defined as the difference in additive genetic mean of the selected top individuals, expected to be parents of the next generation, and the total population after integrating genetic trends out of the posterior distribution of selection responses. Inferences were based on marginal posterior distributions obtained from the Bayesian method for integration over unknown population parameters and “fixed” environmental effects and for appropriate handling of ratio traits. Terminal line pigs (n = 3724) were used for a multi-variate model for ADFI, ADG, and LMP. RFI was estimated from the conditional distribution of ADFI given ADG and LMP, using either genetic (RFIG) or phenotypic (RFIP) partial regression coefficients. The posterior distribution of the FCR’s breeding values was derived from the posterior distribution of “fixed” environmental effects and additive genetic effects on ADFI and ADG.ResultsPosterior means of heritability were 0.32, 0.26, 0.56, 0.20, and 0.15 for ADFI, ADG, LMP, RFIP, and RFIG, respectively. Selection against RFIG showed a direct response of − 0.16 kg/d and correlated responses of − 0.16 kg/kg for FCR and − 0.15 kg/d for ADFI, with no effect on other production traits. Selection against FCR resulted in a direct response of − 0.17 kg/kg and correlated responses of − 0.14 kg/d for RFIG, − 0.18 kg/d for ADFI, and 0.98% for LMP.ConclusionsThe Bayesian methodology developed here enables prediction of breeding values for FCR and RFI from a single multi-variate model. In addition, we derived posterior distributions of direct and correlated responses to selection. Genetic parameter estimates indicated a genetic basis for the studied traits and that genetic improvement through selection was possible. Direct selection against FCR or RFIP resulted in unexpected responses in production traits.

Parameter estimation of multivariate multiple regression model using bayesian with non-informative Jeffreys’ prior distribution

Bayesian method is a method that can be used to estimate the parameters of multivariate multiple regression model. Bayesian method has two distributions, there are prior and posterior distributions. Posterior distribution is influenced by the selection of prior distribution. Jeffreys’ prior distribution is a kind of Non-informative prior distribution. This prior is used when the information about parameter not available. Non-informative Jeffreys’ prior distribution is combined with the sample information resulting the posterior distribution. Posterior distribution is used to estimate the parameter. The purposes of this research is to estimate the parameters of multivariate regression model using Bayesian method with Non-informative Jeffreys’ prior distribution. Based on the results and discussion, parameter estimation of β and Σ which were obtained from expected value of random variable of marginal posterior distribution function. The marginal posterior distributions for β and Σ are multivariate normal and inverse Wishart. However, in calculation of the expected value involving integral of a function which difficult to determine the value. Therefore, approach is needed by generating of random samples according to the posterior distribution characteristics of each parameter using Markov chain Monte Carlo (MCMC) Gibbs sampling algorithm.

Species distribution modeling: a statistical review with focus in spatio-temporal issues

The use of complex statistical models has recently increased substantially in the context of species distribution behavior. This complexity has made the inferential and predictive processes challenging to perform. The Bayesian approach has become a good option to deal with these models due to the ease with which prior information can be incorporated along with the fact that it provides a more realistic and accurate estimation of uncertainty. In this paper, we first review the sources of information and different approaches (frequentist and Bayesian) to model the distribution of a species. We also discuss the Integrated Nested Laplace approximation as a tool with which to obtain marginal posterior distributions of the parameters involved in these models. We finally discuss some important statistical issues that arise when researchers use species data: the presence of a temporal effect (presenting different spatial and spatio-temporal structures), preferential sampling, spatial misalignment, non-stationarity, imperfect detection, and the excess of zeros.

A Note on the Bayesian Modeling of the Stratigraphic Chronology of Canímar Abajo, Cuba

AbstractWe implement a Bayesian statistical analysis of the chronology of Canímar Abajo in Cuba in order to estimate two episodes of burial activity and the period of time corresponding to the hiatus between them. We show that by using simple Bayesian modeling, conclusions can easily be reached by the analysis of the marginal posterior distribution of each parameter of the model. However, we also suggest and describe new statistical tools that exploit the joint posterior distribution of collections of dates. These new tools give complementary information regarding the chronology of human activity.

Forecasting seasonal time series data: a Bayesian model averaging approach

A flexible Bayesian periodic autoregressive model is used for the prediction of quarterly and monthly time series data. As the unknown autoregressive lag order, the occurrence of structural breaks and their respective break dates are common sources of uncertainty these are treated as random quantities within the Bayesian framework. Since no analytical expressions for the corresponding marginal posterior predictive distributions exist a Markov Chain Monte Carlo approach based on data augmentation is proposed. Its performance is demonstrated in Monte Carlo experiments. Instead of resorting to a model selection approach by choosing a particular candidate model for prediction, a forecasting approach based on Bayesian model averaging is used in order to account for model uncertainty and to improve forecasting accuracy. For model diagnosis a Bayesian sign test is introduced to compare the predictive accuracy of different forecasting models in terms of statistical significance. In an empirical application, using monthly unemployment rates of Germany, the performance of the model averaging prediction approach is compared to those of model selected Bayesian and classical (non)periodic time series models.

Phenotypic and genetic relationships between indicators of the mammary gland health status and milk composition, coagulation, and curd firming in dairy sheep

The present study investigated the effect of somatic cell count, lactose, and pH on sheep milk composition, coagulation properties (MCP), and curd firming (CF) parameters. Individual milk samples were collected from 1,114 Sarda ewes reared in 23 farms. Milk composition, somatic cell count, single point MCP (rennet coagulation time, RCT; curd firming time, k20; and curd firmness, a30, a45, and a60), and CF model parameters were achieved. Phenotypic traits were statistically analyzed using a mixed model to estimate the effects of the different levels of milk somatic cell score (SCS), lactose, and pH, respectively. Additive genetic, herd, and residual correlations among these 3 traits, and with milk composition, MCP and CF parameters, were inferred using a Bayesian approach. From a phenotypic point of view, higher SCS levels caused a delayed gelification of milk. Lactose concentration and pH were significant for many milk quality traits, with a very intense effect on both coagulation times and curd firming. These traits (RCT, RCT estimated using the curd firming over time equation, and k20) showed an unfavorable increase of about 20% from the highest to the lowest level of lactose. Milk samples with pH values lower than 6.56 versus higher than 6.78 were characterized by an increase of RCT (from 6.00 to 14.3 min) and k20 (from 1.65 to 2.65 min) and a decrease of all the 3 curd firmness traits. From a genetic point of view, the marginal posterior distribution of heritability estimates evidenced a large and exploitable variability for all 3 phenotypes. The mean intra-farm heritability estimates were 0.173 for SCS, 0.418 for lactose content, and 0.206 for pH. Lactose (favorably), and SCS and pH (unfavorably), at phenotypic and genetic levels, were correlated mainly with RCT and RCT estimated using the curd firming over time equation and scarcely with the other curd firming traits. The SCS, lactose, and pH were significantly correlated with each other's. In conclusion, results reported in the present study suggest that SCS, pH, and lactose affect, contemporarily and independently, milk quality and MCP. These phenotypes, easily available during milk recording schemes measured by infrared spectra prediction, could be used as potential indicators traits for improving cheese-making ability of ovine milk.