Prior Weights Research Articles

503 Separating the Associations of Programming and Tracking during Infancy with Childhood Blood Pressure, the Abcd Study

Background and AimsIn a prospective pregnancy cohort study, we investigated the association of early growth with blood pressure at 5–6 years of age.MethodsOur study was based on 2,338 children who...

1443 Relation of Birth Weight and Infant Growth With Body Size, Body Composition and Fat Distribution at Age 5–6 Years

Background and AimsIn a prospective cohort study (the ABCD study), we investigated the association of early growth with (precursors of) childhood obesity.MethodsOur study was based on 2,338 term born children...

Orthogonalization of vectors with minimal adjustment

Two transformations are proposed that give orthogonal components with a one-to-one correspondence between the original vectors and the components. The aim is that each component should be close to the vector with which it is paired, orthogonality imposing a constraint. The transformations lead to a variety of new statistical methods, including a unified approach to the identification and diagnosis of collinearities, a method of setting prior weights for Bayesian model averaging, and a means of calculating an upper bound for a multivariate Chebychev inequality. One transformation has the property that duplicating a vector has no effect on the orthogonal components that correspond to nonduplicated vectors, and is determined using a new algorithm that also provides the decomposition of a positive-definite matrix in terms of a diagonal matrix and a correlation matrix. The algorithm is shown to converge to a global optimum.

Clustered Mixture Particle Filter for Underwater Multitarget Tracking in Multistatic Active Sonobuoy Systems

The problem of multitarget tracking in underwater multistatic active sonobuoy systems is challenging because of the large number of false contacts and multiple reflections that reach the receivers. Targeting a robust solution that can track an unknown time-varying number of multiple targets, while keeping continuous tracks even in scenarios with large number of false contacts per ping, a particle filter (PF)-based technique is proposed in this paper. The PF is a nonlinear filtering technique that can accommodate arbitrary sensor characteristics, motion dynamics, and noise distributions. An enhanced version of the PF called the mixture PF is utilized in this paper. While the sampling/importance resampling PF samples from the prior importance density and weights the particles according to the observation likelihood, the mixture PF samples from both importance densities and weights the different groups of particles respectively. The usage of this mixture of importance densities provides better performance and faster convergence to the true targets locations. In order to track an unknown time-varying number of targets, two mixture PFs are used (one for target detection and the other for tracking multiple targets), and a density-based clustering technique. The first filter starts with random uniformly distributed samples over the surveillance area and resets every five pings. Just before the reset, the clustering technique runs to detect the clusters that corresponds to different targets and passes them to the second filter. The performance of the proposed technique is examined and demonstrated by different simulated scenarios and some real datasets from the SEABAR07 trial by the NATO Underwater Research Center.

Adaptive estimation of multivariate functions using conditionally Gaussian tensor-product spline priors

We investigate posterior contraction rates for priors on multivariate functions that are constructed using tensor-product B-spline expansions. We prove that using a hierarchical prior with an appropriate prior distribution on the partition size and Gaussian prior weights on the B-spline coefficients, procedures can be obtained that adapt to the degree of smoothness of the unknown function up to the order of the splines that are used. We take a unified approach including important nonparametric statistical settings like density estimation, regression, and classification.

Association between infant correlates of impulsivity - surgency (extraversion) - and early infant growth

Increasing evidence suggests that rapid postnatal weight gain is associated with increased risks of being overweight or obese later in life and of co-morbidities, such as diabetes, the metabolic syndrome and cardiovascular disease. In children as young as two years of age, as well as in adults, an appetitive system-linked impulsivity trait has been demonstrated to be linked with increased overweight, and postulated to act via increased food intake, through greater responsiveness to food and lower self-inhibitory control skills. In this study, we hypothesized that growth in infancy, a critical window for metabolic programming, would be predicted by measures of infant surgency/extraversion, assessed using the Rothbart Infant Behaviour Questionnaire (revised version). Anthropometry was measured at birth and at 3, 6 and 12 months, and weight gains expressed as increases in standardized scores, allowing for adjustment for gender and age, including gestational age. We used conditional weight (CW), a residual of current weight regressed on prior weights, to represent deviations from expected weight gains, from 0 to 3, 3 to 6 and 6 to 12 months. Controlling for significant sociodemographic correlations, multiple regression analyses showed significant prediction of CWs at 3 months but not of CWs at 6 or 12 months by surgency/extraversion. These pilot findings of association between infant growth, during a critical period, and surgency/extraversion, early correlates of impulsivity, warrant further investigation, to ascertain implications for childhood and later weight and body composition.

Deriving prior distributions for Bayesian models used to achieve adaptive e-learning

This paper presents an approach of achieving adaptive e-learning by probabilistically evaluating a learner based not only on the profile and performance data of the learner but also on the data of previous learners. In this approach, an adaptation rule specification language and a user interface tool are provided to a content author or instructor to define adaptation rules. The defined rules are activated at different stages of processing the learning activities of an activity tree which models a composite learning object. System facilities are also provided for modeling the correlations among data conditions specified in adaptation rules using Bayesian Networks. Bayesian inference requires a prior distribution of a Bayesian model. This prior distribution is automatically derived by using the formulas presented in this paper together with prior probabilities and weights assigned by the content author or instructor. Each new learner’s profile and performance data are used to update the prior distribution, which is then used to evaluate the next new learner. The system thus continues to improve the accuracy of learner evaluation as well as its adaptive capability. This approach enables an e-learning system to make proper adaptation decisions even though a learner’s profile and performance data may be incomplete, inaccurate and/or contradictory.

Which uncertainty? Using expert elicitation and expected value of information to design an adaptive program

Natural resource management is plagued with uncertainty of many kinds, but not all uncertainties are equally important to resolve. The promise of adaptive management is that learning in the short-term will improve management in the long-term; that promise is best kept if the focus of learning is on those uncertainties that most impede achievement of management objectives. In this context, an existing tool of decision analysis, the expected value of perfect information (EVPI), is particularly valuable in identifying the most important uncertainties. Expert elicitation can be used to develop preliminary predictions of management response under a series of hypotheses, as well as prior weights for those hypotheses, and the EVPI can be used to determine how much management could improve if uncertainty was resolved. These methods were applied to management of whooping cranes ( Grus americana), an endangered migratory bird that is being reintroduced in several places in North America. The Eastern Migratory Population of whooping cranes had exhibited almost no successful reproduction through 2009. Several dozen hypotheses can be advanced to explain this failure, and many of them lead to very different management responses. An expert panel articulated the hypotheses, provided prior weights for them, developed potential management strategies, and made predictions about the response of the population to each strategy under each hypothesis. Multi-criteria decision analysis identified a preferred strategy in the face of uncertainty, and analysis of the expected value of information identified how informative each strategy could be. These results provide the foundation for design of an adaptive management program.

SELECTION OF WEIGHTS FOR WEIGHTED MODEL AVERAGING

SummaryWe address the task of choosing prior weights for models that are to be used for weighted model averaging. Models that are very similar should usually be given smaller weights than models that are quite distinct. Otherwise, the importance of a model in the weighted average could be increased by augmenting the set of models with duplicates of the model or virtual duplicates of it. Similarly, the importance of a particular model feature (a certain covariate, say) could be exaggerated by including many models with that feature. Ways of forming a correlation matrix that reflects the similarity between models are suggested. Then, weighting schemes are proposed that assign prior weights to models on the basis of this matrix. The weighting schemes give smaller weights to models that are more highly correlated. Other desirable properties of a weighting scheme are identified, and we examine the extent to which these properties are held by the proposed methods. The weighting schemes are applied to real data, and prior weights, posterior weights and Bayesian model averages are determined. For these data, empirical Bayes methods were used to form the correlation matrices that yield the prior weights. Predictive variances are examined, as empirical Bayes methods can result in unrealistically small variances.

Weight Gain in the First Two Years of Life Is an Important Predictor of Schooling Outcomes in Pooled Analyses from Five Birth Cohorts from Low- and Middle-Income Countries

Schooling predicts better reproductive outcomes, better long-term health, and increased lifetime earnings. We used data from 5 cohorts (Brazil, Guatemala, India, the Philippines, and South Africa) to explore the relative importance of birthweight and postnatal weight gain for schooling in pooled analyses (n = 7945) that used appropriate statistical methods [conditional weight (CW) gain measures that are uncorrelated with prior weights] and controlled for confounding. One SD increase in birthweight, ∼0.5 kg, was associated with 0.21 y more schooling and 8% decreased risk of grade failure. One SD increase in CW gain between 0 and 2 y, ∼0.7 kg, was associated with higher estimates, 0.43 y more schooling, and 12% decreased risk of failure. One SD increase of CW gain between 2 and 4 y, ∼0.9 kg, was associated with only 0.07 y more schooling but not with failure. Also, in children born in the lowest tertile of birthweight, 1 SD increase of CW between 0 and 2 y was associated with 0.52 y more schooling compared with 0.30 y in those in the upper tertile. Relationships with age at school entry were inconsistent. In conclusion, weight gain during the first 2 y of life had the strongest associations with schooling followed by birthweight; weight gain between 2 and 4 y had little relationship to schooling. Catch-up growth in smaller babies benefited schooling. Nutrition interventions aimed at women and children under 2 y are among the key strategies for achieving the millennium development goal of universal primary education by 2015.

Size at birth, weight gain in infancy and childhood, and adult blood pressure in 5 low- and middle-income-country cohorts: when does weight gain matter?

BackgroundPromoting catch-up growth in malnourished children has health benefits, but recent evidence suggests that accelerated child weight gain increases adult chronic disease risk. Objective:We aimed to determine how birth weight (BW) and weight gain to midchildhood relate to blood pressure (BP) in young adults. Design:We pooled data from birth cohorts in Brazil, Guatemala, India, the Philippines, and South Africa. We used conditional weight (CW), a residual of current weight regressed on prior weights, to represent deviations from expected weight gain from 0 to 12, 12 to 24, 24 to 48 mo, and 48 mo to adulthood. Adult BP and risk of prehypertension or hypertension (P/HTN) were modeled before and after adjustment for adult body mass index (BMI) and height. Interactions of CWs with small size-for-gestational age (SGA) at birth were tested. Results:Higher CWs were associated with increased BP and odds of P/HTN, with coefficients proportional to the contribution of each CW to adult BMI. Adjusted for adult height and BMI, no child CW was associated with adult BP, but 1 SD of BW was related to a 0.5-mm Hg lower systolic BP and a 9% lower odds of P/HTN. BW and CW associations with systolic BP and P/HTN were not different between adults born SGA and those with normal BW, but higher CW at 48 mo was associated with higher diastolic BP in those born SGA. Conclusions:Greater weight gain at any age relates to elevated adult BP, but faster weight gains in infancy and young childhood do not pose a higher risk than do gains at other ages.

A Bayesian Approach to Utilizing Prior Data in New Drug Development

In this paper we propose a Bayesian method to combine safety data collected from two separate drug development programs using the same active drug substance but for different indications, formulations, or patient populations. The objective of combining the data across the programs is to better define the level of safety risk associated with the new indication or target population. There may be adverse events (AEs) observed in the new program that represent new safety signals. Our method is to explore the AEs using data from both development programs. Our approach utilizes data collected previously to assist in analyzing safety data from the new program. It is assumed that the frequency of a certain AE follows a distribution with a parameter that characterizes the safety risk level. The parameter is assumed to follow a distribution function. In the Bayesian framework, this distribution function is called a prior distribution in the absence of data and posterior distribution when updated by real data. The key concept behind our method is to use data from the previous program to construct a posterior distribution that will in turn serve as a prior distribution for the new program. The construction of this updated prior down weights data from the previous program to emphasize the new program and thus avoids simple pooling of the data across programs. Such “soft use” of previous information minimizes the potential for undue influence of previous data on the analysis. Data from the new program are used to update the prior distribution and compute the posterior distribution for the new program. Key statistics are then calculated from the posterior distribution to quantify the risk level for the new program. We have tested the proposed approach using data from a real Phase 2 study that was conducted as part of a clinical development program for a new indication of an approved drug. The results indicate that the estimated risk level was affected both by the observed event rates and the extents of exposure across the two development programs. This approach appropriately characterizes the safety profile across the two development programs and properly contextualizes new safety signals from the new program.

Nonparametric Bayesian model selection and averaging

We consider nonparametric Bayesian estimation of a probabil- ity density p based on a random sample of size n from this density using a hierarchical prior. The prior consists, for instance, of prior weights on the regularity of the unknown density combined with priors that are ap- propriate given that the density has this regularity. More generally, the hierarchy consists of prior weights on an abstract model index and a prior on a density model for each model index. We present a general theorem on the rate of contraction of the resulting posterior distribution as n ! 1, which gives conditions under which the rate of contraction is the one at- tached to the model that best approximates the true density of the obser- vations. This shows that, for instance, the posterior distribution can adapt to the smoothness of the underlying density. We also study the posterior distribution of the model index, and find that under the same conditions the posterior distribution gives negligible weight to models that are bigger than the optimal one, and thus selects the optimal model or smaller models that also approximate the true density well. We apply these result to log spline density models, where we show that the prior weights on the regu- larity index interact with the priors on the models, making the exact rates depend in a complicated way on the priors, but also that the rate is fairly robust to specification of the prior weights.

On adaptive Bayesian inference

We study the rate of Bayesian consistency for hierarchical priors consisting of prior weights on a model index set and a prior on a density model for each choice of model index. Ghosal, Lember and Van der Vaart [2] have obtained general in-probability theorems on the rate of convergence of the resulting posterior distributions. We extend their results to almost sure assertions. As an application we study log spline densities with a finite number of models and obtain that the Bayes procedure achieves the optimal minimax rate n−γ/(2γ+1) of convergence if the true density of the observations belongs to the Hölder space Cγ[0,1]. This strengthens a result in [1; 2]. We also study consistency of posterior distributions of the model index and give conditions ensuring that the posterior distributions concentrate their masses near the index of the best model.

A method for calibrated maximum likelihood classification of forest types

A modification to the maximum likelihood algorithm was developed for classification of forest types in Sweden's part of the CORINE land cover mapping project. The new method, called the “calibrated maximum likelihood classification” involves an automated and iterative adjustment of prior weights until class frequency in the output corresponds to class frequency as calculated from objective (field-inventoried) estimates. This modification compensates for the maximum likelihood algorithm's tendency to over-represent dominant classes and under-represent less frequent ones. National forest inventory plot data measured from a five-year period are used to estimate relative frequency of class occurrence and to derive spectral signatures for each forest class. The classification method was implemented operationally within an automated production system which allowed rapid production of a country-wide forest type map from Landsat TM/ETM+ satellite data. The production system automated the retrieval and updating of forest inventory plots, a plot-to-image matching routine, illumination and haze correction of satellite imagery, and classification into forest classes using the calibrated maximum likelihood classification. This paper describes the details of the method and demonstrates the result of using an iterative adjustment of prior weights versus unadjusted prior weights. It shows that the calibrated maximum likelihood algorithm adjusts for the overclassification of classes that are well represented in the training data as well as for other classes, resulting in an output where class proportions are close to those as expected based on forest inventory data.

On universal Bayesian adaptation

We consider estimating a probability density p based on a random sample from this density by a Bayesian approach. The prior is constructed in two steps, by first constructing priors on a collection of models each expressing a qualitative prior guess on the true density, and next combining these priors in an overall prior by attaching prior weights to the models. The purpose is to show that the posterior distribution contracts to the true distribution at a rate that is (nearly) equal to the rate that would have been obtained had only the model that is most suitable for the true density been used. We study special model weights that yield this adaptation property in some generality. Examples include minimal discrete priors and finite-dimensional models, with special attention to scales of Banach spaces, such as Hölder spaces, spline models, and classes of densities that are not uniformly bounded away from zero or infinity.

A Comparative Study of IRT Fixed Parameter Calibration Methods

This article provides technical descriptions of five fixed parameter calibration (FPC) methods, which were based on marginal maximum likelihood estimation via the EM algorithm, and evaluates them through simulation. The five FPC methods described are distinguished from each other by how many times they update the prior ability distribution and by how many EM cycles they use. Specifically, the five FPC methods included no prior weights updating and one EM cycle (NWU‐OEM) or multiple EM cycles (NWU‐MEM), one prior weights updating and one EM cycle (OWU‐OEM) or multiple EM cycles (OWU‐MEM), and multiple weights updating and multiple EM cycles (MWU‐MEM) methods. All the five FPC methods were evaluated in terms of recovery of the underlying ability distribution and item parameters. An important factor in the simulation was three different ability (normal) distributions—N(0, 1), N(0.5, 1.22), and N(1, 1.42)—for FPC groups, with the fixed item parameters obtained with a reference N(0, 1) group. Only the MWU‐MEM method appeared to perform properly under all the three distributions. Under the N(0, 1) distribution, the NWU‐MEM and OWU‐MEM methods also appeared to perform properly. Under the N(0.5, 1.22), and N(1, 1.42) distributions, however, the four methods other than the MWU‐MEM method resulted in some or severe under‐estimation in the recovery.

MODEL WEIGHTS AND THE FOUNDATIONS OF MULTIMODEL INFERENCE

Statistical thinking in wildlife biology and ecology has been profoundly influenced by the introduction of AIC (Akaike's information criterion) as a tool for model selection and as a basis for model averaging. In this paper, we advocate the Bayesian paradigm as a broader framework for multimodel inference, one in which model averaging and model selection are naturally linked, and in which the performance of AIC-based tools is naturally evaluated. Prior model weights implicitly associated with the use of AIC are seen to highly favor complex models: in some cases, all but the most highly parameterized models in the model set are virtually ignored a priori. We suggest the usefulness of the weighted BIC (Bayesian information criterion) as a computationally simple alternative to AIC, based on explicit selection of prior model probabilities rather than acceptance of default priors associated with AIC. We note, however, that both procedures are only approximate to the use of exact Bayes factors. We discuss and illustrate technical difficulties associated with Bayes factors, and suggest approaches to avoiding these difficulties in the context of model selection for a logistic regression. Our example highlights the predisposition of AIC weighting to favor complex models and suggests a need for caution in using the BIC for computing approximate posterior model weights.

Variance Component Estimation in Linear Inverse Ill-posed Models

Regularization has been applied by implicitly assuming that the weight matrix of measurements is known. If measurements are assumed to be heteroscedastic with different unknown variance components, all regularization techniques may not be proper to apply, unless techniques of variance component estimation are directly implemented. Although variance component estimation techniques have been proposed to simultaneously estimate the variance components and provide a means of regularization, the regularization parameter is treated as if it were also an extra variance component. In this paper, we assume no prior information on the model parameters and do not treat the regularization parameter as an extra variance component. Instead, we first analyze the biases of estimated variance components due to the regularization parameter and then propose bias-corrected variance component estimators. The results have shown that they work very well. Finally, we propose and investigate through simulations an iterative scheme to simultaneously estimate the variance components and the regularization parameter, in order to eliminate the effect of regularization parameter on variance components and the effect of incorrect prior weights or initial variance components on the regularization parameter.

Under the skin: On the impartial treatment of genetic and environmental hypotheses of racial differences.

Environmental and genetic explanations have been given for Black-White racial differences in intelligence and other traits. In science, viable, alternative hypotheses are ideally given equal Bayesian prior weights; but this has not been true in the study of racial differences. This article advocates testing environmental and genetic hypotheses of racial differences as competing hypotheses. Two methods are described: (a) fitting means within structural equation models and (b) predicting means of interracial children. These methods have limitations that call for improved research designs of racial differences. One improvement capitalizes on biotechnology. Genetic admixture estimates--the percentage of genes of European origin that a Black individual possesses (independent of genes related to skin coloration)--can represent genetic influences. The study of interracial children can be improved by increasing sample size and by choosing family members who are most informative for a research question. Eventually, individual-admixture estimates will be replaced by molecular genetic tests of alleles of those genes that influence traits.