Highest Posterior Density Regions Research Articles

Mixed Model Studies on Inheritance of Reproductive Traits in Laying Hens – A Bayesian Approach

Segregation analyses performed for many livestock species indicate a mixed inheritance model of reproductive traits. Additionally, depending on the population, a given trait can be determined by a number of genes with large effects. Genetic backgrounds of hatch-ability and fertility in poultry are still not known sufficiently. The objectives of this study are to verify the hypothesis on segregation of single genes (1 vs. 2) affecting fertility and hatchability and to estimate a heritability of these traits. Records from 2,040 and 2,015 dams from full-pedigreed strains of Rhode Island Red (R33) and New Hampshire (N88) from a pedigree farm were analyzed. The percentage of fertilized eggs and the percentage of the eggs hatched of fertilized eggs were registered for dams only. Fertility was checked by candling on the eighth day of incubation. To obtain a binomial phenotypic scale, 10 eggs per dam were included into the analysis. Animal single-trait threshold models were used for the analysis of data. The first model included the effects of 2 single genes, 2 fixed effects of year and season, additive polygenic effects, and permanent environmental effects. In the second model, only 1 single gene effect was included. Additionally, the analysis based on the polygenic threshold model was also performed. The Gibbs sampling procedure was used. The significance of single gene effects was verified by highest posterior density regions. The obtained results clearly gave evidence for the segregation of 1 major gene for hatchability in strain R33. Furthermore, the mixed inheritance model can also be suggested for fertility in this strain. After the analysis, the polygenic heritabilities were very low (<0.11), whereas major polygenic heritability ranged from 0.05 to 0.12.

Empirical Bayes factor analyses of quantitative trait loci for gestation length in Iberian × Meishan F2 sows

The aim of this study was to investigate chromosomal regions affecting gestation length in sows. An experimental F2 cross between Iberian and Meishan pig breeds was used for this purpose and we genotyped 119 markers covering the 18 porcine autosomal chromosomes. Within this context, we have developed a new empirical Bayes factor (BF) approach to compare between nested models, with and without the quantitative trait loci (QTL) effect, and after including the location of the QTL as an unknown parameter in the model. This empirical BF can be easily calculated from the output of a Markov chain Monte Carlo sampling by averaging conditional densities at the null QTL effects. Linkage analyses were performed in each chromosome using an animal model to account for infinitesimal genetic effects. Initially, three QTL were detected at chromosomes 6, 8 and 11 although, after correcting for multiple testing, only the additive QTL located in cM 110 of chromosome 8 remained. For this QTL, the allelic effect of substitution of the Iberian allele increased gestation length in 0.521 days, with a highest posterior density region at 95% ranged between 0.121 and 0.972 days. Although future studies are necessary to confirm if detected QTL is relevant and segregating in commercial pig populations, a hot-spot on the genetic regulation of gestation length in pigs seems to be located in chromosome 8.

Investigation of major gene for milk yield, milking speed, dry matter intake, and body weight in dairy cattle

The main aim of this study was to determine if there exist any major gene for milk yield (MY), milking speed (MS), dry matter intake (DMI), and body weight (BW) recorded at various stages of lactation in first-lactation dairy cows (2543 observations from 320 cows) kept at the research farm of the Swiss Federal Institute of Technology between April 1994 and April 2004. Data were modelled based a simple repeatability covariance structure and analysed by using Bayesian segregation analyses. Gibbs sampling was used to make statistical inferences on posterior distributions; inferences were based on a single run of the Markov chain for each trait with 500,000 samples, with each 10th sample collected because of the high correlation among the samples. The posterior mean (+/-SD) of major gene variance was 2.61 (+/-2.46) for MY, 0.83 (+/-1.26) for MS, 4.37 (+/-2.34) for DMI, and 2056.43 (+/-665.67) for BW. Highest posterior density regions for 3 of the 4 traits did not include 0 (except MS), which supported the evidence for major gene. With additional tests for agreement with Mendelian transmission probabilities, we could only confirm the existence of a major gene for MY, but not for MS, DMI, and BW. Expected Mendelian transmission probabilities and their model fits were also compared.

Testing genetic determinism in rate of hoof growth in pigs using Bayes Factors

Abstract Excessive growth and consequent deformity of hooves is a frequent disorder in some purebred pig populations. A test to detect possible genetic determinism related with this phenomenon was performed using the Bayes Factor (BF). Data were available for females from three purebred selection lines: Landrace (561 records), Pietrain (183) and Large White (225). Animals were scored in four categories, according to the overall growth rate of their hooves. A Bayesian analysis was performed separately for each line using a threshold model with a probit approach, and Bayes Factors between models with and without additive genetic effects were computed. Results from the three lines showed that models exhibiting genetic variability were much more probable than those that did not include a genetic component, with BF values of 312, 35 and 40 (and posterior probabilities of 0.99, 0.97 and 0.98), respectively, for the Landrace, Pietrain and Large White lines. Monte Carlo estimates of posterior means of heritabilities were medium to high (0.25, 0.41 and 0.38, respectively), and the highest posterior density region for heritability at 99% did not include zero in any of the three lines. These results allow us to conclude that genetic determinism has an important influence upon the rate of hoof growth in the pig. A potential genetic response can be achieved in the populations analysed, but further studies are needed to determine the genetic architecture of hoof growth disorders in pigs.

Incorporating δ18O values of past waters in the calibration of radiocarbon dating

All scientists who study the late Quaternary are confronted by the problem of radiocarbon calibration. Since 1969, numerous high-frequency time series have been developed through the use of several stable isotopes (e.g., δ 1 8 O). Such time series are inefficient as dating methods because too many age possibilities are obtained for a given unique value. However, the Bayesian framework permits incorporation of these time series as a prior assumption on ages in order to improve calibrated ages obtained from a more precise absolute dating approach such as that using radiocarbon. The method is tested on data obtained from the Gigny cave (French Jura). Highly discontinuous posterior distributions are obtained with narrow highest posterior density regions. Thus, when the radiocarbon method alone gives large uncertainties, the incorporation of δ 1 8 O values of meteoric waters largely diminishes these uncertainties.

Bayesian analysis of complementary Poisson rate parameters with data subject to misclassification

We formulate closed-form Bayesian estimators for two complementary Poisson rate parameters using double sampling with data subject to misclassification and error free data. We also derive closed-form Bayesian estimators for two misclassification parameters in the modified Poisson model we assume. We use our results to determine credible sets for the rate and misclassification parameters. Additionally, we use MCMC methods to determine Bayesian estimators for three or more rate parameters and the misclassification parameters. We also perform a limited Monte Carlo simulation to examine the characteristics of these estimators. We demonstrate the efficacy of the new Bayesian estimators and highest posterior density regions with examples using two real data sets.

Simultaneous Posterior Probability Statements From Monte Carlo Output

This article considers the problem of making simultaneous probability statements in multivariate inferential problems based on samples from a posterior distribution. The calculation of simultaneous credible bands is reviewed and—as an alternative—contour probabilities are proposed. These are defined as 1 minus the content of the highest posterior density region which just covers a certain point of interest. We discuss a Monte Carlo method to estimate contour probabilities and distinguish whether or not the functional form of the posterior density is available. In the latter case, an approach based on Rao-Blackwellization is proposed. We highlight that this new estimate has an important invariance property. We illustrate the performance of the different methods in three applications.

Uncertainty in most probable number calculations for microbiological assays.

Microbiological assays commonly use incubations of multiple tubes in a dilution series, and microorganism concentration is read as a most probable number (MPN) in standard tables for the observed pattern of positive tubes. Published MPN tables differ, sometimes substantially, because of use of approximate MPN calculation procedures, different rounding conventions in the results, and different methods of calculating confidence or credible intervals. We conclude that the first 2 issues can now be resolved by using recently developed exact MPN calculation methods and by reporting rounding conventions in standard tables. The third issue is not amenable to complete resolution, especially if credible interval (as opposed to confidence interval) limits are desired--as we think they most often are. In that case, Bayesian statistics are called for and the analyst must provide a distribution of concentration that was presumed to be true before the assay was performed. This is mathematically combined with the assay data, resulting in a posterior concentration distribution. These distributions may then be used to quantify the uncertainty in the MPN estimate, and the best approach is to use the highest posterior density regions of these distributions. If based on diffuse prior information (positing that, prior to an assay being performed, all positive concentrations are equally likely), then established procedures might be used to calculate the limits and publish them in standard tables. In the event that this prior assumption is held to be not satisfactory, we show results for an empirical Bayes procedure, with a Poisson prior distribution, giving credible interval widths much narrower than in the other cases examined.

Visual assessment of post-mortem lesions exhibits little additive genetic variation in growing pigs

This study tested the premise that additive genetic variation for incidence of post-mortem disease lesions visually assessed during routine carcass inspection exists in growing pigs. A total of 23 478 male growing pigs were assessed for three categories of disease lesions: any post-mortem lesion; respiratory lesions; and abscesses. Additive genetic variation for each category of lesions was estimated by fitting a categorical threshold liability model, formulated in a Bayesian context, to the incidences of the lesions (i.e. not diagnosed/diagnosed with one lesion/diagnosed with two or more different lesions). The amount of additive genetic variation detected for the incidence of any post-mortem lesion on the underlying liability scale was very low [additive genetic variance (marginal posterior mean, 95% highest posterior density region)=0.019, 0.000–0.046; heritability (marginal posterior mean, 95% highest posterior density region)=0.03, 0.00–0.06]. By comparison, the amount of additive genetic variation was negligible for respiratory lesions (0.002, 0.000–0.015; 0.00, 0.00–0.03) and low for abscesses (0.050, 0.001–0.098; 0.08, 0.00–0.15). The general lack of additive genetic variation suggests that the incidence of post-mortem lesions visually assessed during routine carcass inspection would not provide a suitable criterion by which to select pigs for resistance to clinical and sub-clinical disease.

Multivariate analysis of litter size for multiple parities with production traits in pigs: II. Response to selection for litter size and correlated response to production traits1

Litter size and production trait responses to experimental selection for increased litter size in a Landrace pig population are reported. The numbers of sows and litters available for the first cycle of selection were 3,034 and 961, respectively. Selection was carried out using a BLUP repeatability animal model for number of piglets born alive (NBA). The experiment included one selection and one control line, each with three nonoverlapping generations. The selection line (H) consisted of the 160 sows with the highest breeding values and one boar from each of 25 full-sib families with the highest breeding values. The control line (C) consisted of 160 sows and 25 boars randomly chosen. The two subsequent generations in each line were obtained by random selection. A Bayesian analysis of genetic response using a multivariate model was carried out by Gibbs sampler. Marginal posterior distributions were obtained for direct response in NBA, and for correlated response in weight (WT), and backfat thickness (BT) at 175 d of age. The posterior means and posterior standard deviation (PSD) for direct genetic response of NBA ranged from 0.32 (PSD 0.08) in the first parity to 0.64 (PSD 0.08) in the fourth. The posterior means for correlated genetic response in WT and BT were −0.66 kg (PSD 0.36) and 0.20 mm (PSD 0.10), respectively. For WT and BT, the 95% highest posterior density regions (HPD) contain zero-correlated genetic response. Marginal posterior distributions of selection differentials were investigated. The posterior means for standardized selection differentials for NBA in different parities ranged from 0.70 (PSD 0.12) to 0.94 (PSD 0.06) in females for line H, from 0.22 (PSD 0.19) to 0.34 (PSD 0.10) in males for line H, and from 0.08 (PSD 0.08) to 0.13 (PSD 0.07) in females for line C. All available males were used in line C. Results from this experiment showed that selection for increased litter size is effective. Responses to selection were heterogeneous across parities, suggesting that litter size in each parity may have a different genetic background. No correlated genetic response to growth and backfat thickness was observed.

An optimal two-stage graphical search planning procedure for submerged targets

The graphical methods presented by Koopman [1,2] in the classical stationary search theory are extended to a two-stage search of a submerged target, subject to an overall budgetary constraint and with possible budget transfer between the stages. An algorithm, based on the technique for computing the highest posterior density region (hpd) in Bayesian statistics, developed by the authors, and an associated computer code, are provided to permit a convenient and precise graphical solution to the problem, instead of using the usual analytic approach. A numerical example, formulated under realistic hypotheses, illustrates the various capabilities of this code.

Multivariate analysis of litter size for multiple parities with production traits in pigs: II. Response to selection for litter size and correlated response to production traits.

Litter size and production trait responses to experimental selection for increased litter size in a Landrace pig population are reported. The numbers of sows and litters available for the first cycle of selection were 3,034 and 961, respectively. Selection was carried out using a BLUP repeatability animal model for number of piglets born alive (NBA). The experiment included one selection and one control line, each with three nonoverlapping generations. The selection line (H) consisted of the 160 sows with the highest breeding values and one boar from each of 25 full-sib families with the highest breeding values. The control line (C) consisted of 160 sows and 25 boars randomly chosen. The two subsequent generations in each line were obtained by random selection. A Bayesian analysis of genetic response using a multivariate model was carried out by Gibbs sampler. Marginal posterior distributions were obtained for direct response in NBA, and for correlated response in weight (WT), and backfat thickness (BT) at 175 d of age. The posterior means and posterior standard deviation (PSD) for direct genetic response of NBA ranged from 0.32 (PSD 0.08) in the first parity to 0.64 (PSD 0.08) in the fourth. The posterior means for correlated genetic response in WT and BT were -0.66 kg (PSD 0.36) and 0.20 mm (PSD 0.10), respectively. For WT and BT, the 95% highest posterior density regions (HPD) contain zero-correlated genetic response. Marginal posterior distributions of selection differentials were investigated. The posterior means for standardized selection differentials for NBA in different parities ranged from 0.70 (PSD 0.12) to 0.94 (PSD 0.06) in females for line H, from 0.22 (PSD 0.19) to 0.34 (PSD 0.10) in males for line H, and from 0.08 (PSD 0.08) to 0.13 (PSD 0.07) in females for line C. All available males were used in line C. Results from this experiment showed that selection for increased litter size is effective. Responses to selection were heterogeneous across parities, suggesting that litter size in each parity may have a different genetic background. No correlated genetic response to growth and backfat thickness was observed.

Confidence regions for GPS baselines by Bayesian statistics

The global positioning system (GPS) model is distinctive in the way that the unknown parameters are not only real-valued, the baseline coordinates, but also integers, the phase ambiguities. The GPS model therefore leads to a mixed integer–real-valued estimation problem. Common solutions are the float solution, which ignores the ambiguities being integers, or the fixed solution, where the ambiguities are estimated as integers and then are fixed. Confidence regions, so-called HPD (highest posterior density) regions, for the GPS baselines are derived by Bayesian statistics. They take care of the integer character of the phase ambiguities but still consider them as unknown parameters. Estimating these confidence regions leads to a numerical integration problem which is solved by Monte Carlo methods. This is computationally expensive so that approximations of the confidence regions are also developed. In an example it is shown that for a high confidence level the confidence region consists of more than one region.

Some new Results on Probability Matching Priors

The paper has three components. First, for a realvalued parameter of interest orthogonal (Cox and Reid, 1987) to the nuisance parameter vector, we find a necessary and sufficient condition for the equivalence of second order quantile matching priors and highest posterior density regions matching priors within the class of first order quantile matching priors. Examples are presented to illustrate the result. Second, we develop a quantile matching prior in a normal hierarchical Bayesian model. This prior turns out to be different from the one proposed earlier by Morris (1983). Third, we obtain an exact matching result when the objective is prediction of a real-valued random variable from a location family of distributions. AMS (2000) Subject Classification: 62F15, 62F25, 62E20

On Confidence Intervals Associated With the Usual and Adjusted Likelihoods

Summary In the presence of nuisance parameters, we drive an explicit higher order asymptotic formula to compare the expected lengths of confidence intervals given by likelihood ratio statistics arising from the usual profile likelihood and various adjustments thereof. Highest posterior density regions, with approximate frequentist validity, are also included in the study.

Evidence and Credibility: Full Bayesian Significance Test for Precise Hypotheses

A Bayesian measure of evidence for precise hypotheses is presented. The intention is to give a Bayesian alternative to significance tests or, equivalently, to p-values. In fact, a set is defined in the parameter space and the posterior probability, its credibility, is evaluated. This set is the "Highest Posterior Density Region" that is "tangent" to the set that defines the null hypothesis. Our measure of evidence is the complement of the credibility of the "tangent" region.

Interval-based versus decision theoretic criteria for the choice of sample size

SUMMARY Several different criteria for Bayesian sample size determination have recently been proposed. Bayesian approaches are natural, since at the planning stage of an experiment one is forced to consider prior notions about unknown parameter values that may affect the choice of a final sample size. For this, all the methods consider a prior distribution over the unknown parameters. Differences between the methods have been driven by the type of inferences that will be made, e.g. hypothesis testing or interval estimation, the latter based on posterior means and variances or highest posterior density regions. A more fundamental question, however, is whether to introduce formally a loss or utility function to aid in choosing the sample size. In this paper, we discuss the advantages and disadvantages of taking a fully decision theoretic approach versus one of the simpler approaches, which only implicitly consider utilities in balancing increased precision against the increased costs associated with larger sample sizes. Throughout, we emphasize the practical aspects of sample size estimation, raising issues that would face the consumer of statistics in selecting a sample size in a given experiment.

Bayesian ikference procedures derived via the concept of relative surprise

We consider the problem of deriving Bayesian inference procedures via the concept of relative surprise. The mathematical concept of surprise has been developed by I.J. Good in a long sequence of papers. We make a modification to this development that permits the avoidance of a serious defect; namely, the change of variable problem. We apply relative surprise to the development of estimation, hypothesis testing and model checking procedures. Important advantages of the relative surprise approach to inference include the lack of dependence on a particular loss function and complete freedom to the statistician in the choice of prior for hypothesis testing problems. Links are established with common Bayesian inference procedures such as highest posterior density regions, modal estimates and Bayes factors. From a practical perspective new inference procedures arise that possess good properties.

A Bayesian growth and yield model for slash pine plantations

We formulate a traditional growth and yield model as a Bayes model. We attempt to introduce as few new assumptions as possible. Zellner's Bayesian method of moments procedure is used, because the published model did not include any distributional assumptions. We generate predictive posterior samples for a number of stand variables using the Gibbs sampler. The means of the samples compare favorably with the predictions from the published model. In addition, our model delivers distributions of outcomes, from which it is easy to establish measures of uncertainty, such as highest posterior density regions.

Bayesian inference, Gibbs' sampler and uncertainty estimation in geophysical inversion1

AbstractThe posterior probability density function (PPD), σ(m|dobs), of earth model m, where dobs are the measured data, describes the solution of a geophysical inverse problem, when a Bayesian inference model is used to describe the problem. In many applications, the PPD is neither analytically tractable nor easily approximated and simple analytic expressions for the mean and variance of the PPD are not available. Since the complete description of the PPD is impossible in the highly multi‐dimensional model space of many geophysical applications, several measures such as the highest posterior density regions, marginal PPD and several orders of moments are often used to describe the solutions. Calculation of such quantities requires evaluation of multidimensional integrals. A faster alternative to enumeration and blind Monte‐Carlo integration is importance sampling which may be useful in several applications. Thus how to draw samples of m from the PPD becomes an important aspect of geophysical inversion such that importance sampling can be used in the evaluation of these multi‐dimensional integrals. Importance sampling can be carried out most efficiently by a Gibbs' sampler (GS). We also introduce a method which we called parallel Gibbs' sampler (PGS) based on genetic algorithms (GA) and show numerically that the results from the two samplers are nearly identical.We first investigate the performance of enumeration and several sampling based techniques such as a GS, PGS and several multiple maximum a posteriori (MAP) algorithms for a simple geophysical problem of inversion of resistivity sounding data. Several non‐linear optimization methods based on simulated annealing (SA), GA and some of their variants can be devised which can be made to reach very close to the maximum of the PPD. Such MAP estimation algorithms also sample different points in the model space. By repeating these MAP inversions several times, it is possible to sample adequately the most significant portion(s) of the PPD and all these models can be used to construct the marginal PPD, mean) covariance, etc. We observe that the GS and PGS results are identical and indistinguishable from the enumeration scheme. Multiple MAP algorithms slightly underestimate the posterior variances although the correlation values obtained by all the methods agree very well. Multiple MAP estimation required 0.3% of the computational effort of enumeration and 40% of the effort of a GS or PGS for this problem. Next, we apply GS to the inversion of a marine seismic data set to quantify uncertainties in the derived model, given the prior distribution determined from several common midpoint gathers.