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PubMed2006

Structural inference in transition measurement error models for longitudinal data.

We propose a new class of models, transition measurement error models, to study the effects of covariates and the past responses on the current response in longitudinal studies when one of the covariates is measured with error. We show that the response variable conditional on the error-prone covariate follows a complex transition mixed effects model. The naive model obtained by ignoring the measurement error correctly specifies the transition part of the model, but misspecifies the covariate effect structure and ignores the random effects. We next study the asymptotic bias in naive estimator obtained by ignoring the measurement error for both continuous and discrete outcomes. We show that the naive estimator of the regression coefficient of the error-prone covariate is attenuated, while the naive estimators of the regression coefficients of the past responses are generally inflated. We then develop a structural modeling approach for parameter estimation using the maximum likelihood estimation method. In view of the multidimensional integration required by full maximum likelihood estimation, an EM algorithm is developed to calculate maximum likelihood estimators, in which Monte Carlo simulations are used to evaluate the conditional expectations in the E-step. We evaluate the performance of the proposed method through a simulation study and apply it to a longitudinal social support study for elderly women with heart disease. An additional simulation study shows that the Bayesian information criterion (BIC) performs well in choosing the correct transition orders of the models.

Aged↗
PubMed1999

Bayesian inference applied to the electromagnetic inverse problem.

We present a new approach to the electromagnetic inverse problem that explicitly addresses the ambiguity associated with its ill-posed character. Rather than calculating a single "best" solution according to some criterion, our approach produces a large number of likely solutions that both fit the data and any prior information that is used. Whereas the range of the different likely results is representative of the ambiguity in the inverse problem even with prior information present, features that are common across a large number of the different solutions can be identified and are associated with a high degree of probability. This approach is implemented and quantified within the formalism of Bayesian inference, which combines prior information with that of measurement in a common framework using a single measure. To demonstrate this approach, a general neural activation model is constructed that includes a variable number of extended regions of activation and can incorporate a great deal of prior information on neural current such as information on location, orientation, strength, and spatial smoothness. Taken together, this activation model and the Bayesian inferential approach yield estimates of the probability distributions for the number, location, and extent of active regions. Both simulated MEG data and data from a visual evoked response experiment are used to demonstrate the capabilities of this approach.

Bayes Theorem↗
PubMed2003

Performance-based selection of likelihood models for phylogeny estimation.

Phylogenetic estimation has largely come to rely on explicitly model-based methods. This approach requires that a model be chosen and that that choice be justified. To date, justification has largely been accomplished through use of likelihood-ratio tests (LRTs) to assess the relative fit of a nested series of reversible models. While this approach certainly represents an important advance over arbitrary model selection, the best fit of a series of models may not always provide the most reliable phylogenetic estimates for finite real data sets, where all available models are surely incorrect. Here, we develop a novel approach to model selection, which is based on the Bayesian information criterion, but incorporates relative branch-length error as a performance measure in a decision theory (DT) framework. This DT method includes a penalty for overfitting, is applicable prior to running extensive analyses, and simultaneously compares all models being considered and thus does not rely on a series of pairwise comparisons of models to traverse model space. We evaluate this method by examining four real data sets and by using those data sets to define simulation conditions. In the real data sets, the DT method selects the same or simpler models than conventional LRTs. In order to lend generality to the simulations, codon-based models (with parameters estimated from the real data sets) were used to generate simulated data sets, which are therefore more complex than any of the models we evaluate. On average, the DT method selects models that are simpler than those chosen by conventional LRTs. Nevertheless, these simpler models provide estimates of branch lengths that are more accurate both in terms of relative error and absolute error than those derived using the more complex (yet still wrong) models chosen by conventional LRTs. This method is available in a program called DT-ModSel.

Bayes Theorem↗
PubMed2006

Probabilistic mechanisms in sensorimotor control.

Uncertainty constitutes a fundamental constraint on human sensorimotor control. Our sensors are noisy and do not provide perfect information about all the properties of the world. Moreover, our muscles generate noisy outputs and many tasks we perform vary in an unpredictable way. Here we review the computations that the CNS uses in the face of such sensory, motor and task uncertainty. We show that the CNS reduces the uncertainty in estimates about the state of the world by using a Bayesian combination of prior knowledge and sensory feedback. It is shown that these mechanisms generalize to state estimation of ones own body during movement. We review how the CNS optimizes decisions based on these estimates, examining the error criterion that people optimize when performing targeted movements. Finally, we describe how signal-dependent noise on the motor output places constraints on performance. Goal-directed movement arises from a model in which the statistics of our actions are optimized. Together these studies provide a probabilistic framework for sensorimotor control.

Animals↗
PubMedDate not supplied

Automatic basis selection techniques for RBF networks.

This paper proposes a generic criterion that defines the optimum number of basis functions for radial basis function (RBF) neural networks. The generalization performance of an RBF network relates to its prediction capability on independent test data. This performance gives a measure of the quality of the chosen model. An RBF network with an overly restricted basis gives poor predictions on new data, since the model has too little flexibility (yielding high bias and low variance). By contrast, an RBF network with too many basis functions also gives poor generalization performance since it is too flexible and fits too much of the noise on the training data (yielding low bias but high variance). Bias and variance are complementary quantities, and it is necessary to assign the number of basis function optimally in order to achieve the best compromise between them. In this paper we use Stein's unbiased risk estimator to derive an analytical criterion for assigning the appropriate number of basis functions. Two cases of known and unknown noise have been considered and the efficacy of this criterion in both situations is illustrated experimentally. The paper also shows an empirical comparison between this method and two well known classical methods, cross validation and the Bayesian information criterion, BIC.

Electronic Data Processing↗
PubMed2005

Bayesian analysis for generalized linear models with nonignorably missing covariates.

We propose Bayesian methods for estimating parameters in generalized linear models (GLMs) with nonignorably missing covariate data. We show that when improper uniform priors are used for the regression coefficients, phi, of the multinomial selection model for the missing data mechanism, the resulting joint posterior will always be improper if (i) all missing covariates are discrete and an intercept is included in the selection model for the missing data mechanism, or (ii) at least one of the covariates is continuous and unbounded. This impropriety will result regardless of whether proper or improper priors are specified for the regression parameters, beta, of the GLM or the parameters, alpha, of the covariate distribution. To overcome this problem, we propose a novel class of proper priors for the regression coefficients, phi, in the selection model for the missing data mechanism. These priors are robust and computationally attractive in the sense that inferences about beta are not sensitive to the choice of the hyperparameters of the prior for phi and they facilitate a Gibbs sampling scheme that leads to accelerated convergence. In addition, we extend the model assessment criterion of Chen, Dey, and Ibrahim (2004a, Biometrika 91, 45-63), called the weighted L measure, to GLMs and missing data problems as well as extend the deviance information criterion (DIC) of Spiegelhalter et al. (2002, Journal of the Royal Statistical Society B 64, 583-639) for assessing whether the missing data mechanism is ignorable or nonignorable. A novel Markov chain Monte Carlo sampling algorithm is also developed for carrying out posterior computation. Several simulations are given to investigate the performance of the proposed Bayesian criteria as well as the sensitivity of the prior specification. Real datasets from a melanoma cancer clinical trial and a liver cancer study are presented to further illustrate the proposed methods.

Bayes Theorem↗
PubMed2007

Estimation of genetic parameters for cumulative egg numbers in a broiler dam line by using a random regression model.

The random regression model (RRM) methodology was applied to the estimation of genetic parameters for cumulative egg numbers and monthly egg production in a broiler dam line. The data were extracted from records of a commercial dam line in 2001 to 2003. A total of 99,193 records from 6,475 hens and 9,111 pedigreed animals were used in the current study. The variance components were estimated using Gibbs sampling procedure. According to the Bayesian information criterion and Bayes factor, an RRM with Legendre polynomial of 2 orders for hatching groups and additive genetic effects and of 4 orders for permanent environmental effects was chosen as the optimal model for cumulative egg numbers in the broiler dam line. The heritability estimates of the cumulative egg numbers between wk 1 and 40 of production ranged from 0.16 to 0.54, whereas heritability estimates from wk 12 to 20 of production were moderate. The ratios of permanent environmental variance to phenotypic variance were large, indicating that the RRM could produce better estimates of additive genetic effects. The genetic and phenotypic correlations between cumulative egg numbers at different production weeks estimated with the optimal RRM were generally higher when the overlapping weeks were greater. In addition, genetic parameters for monthly egg production could also be obtained by the optimal RRM, and the heritability estimates ranged from 0.03 to 0.18. It was suggested that early selection based on cumulative egg numbers in the first 19 wk of production could effectively improve annual egg production in the broiler dam line.

Animals↗
PubMed2006

The time-profile of cell growth in fission yeast: model selection criteria favoring bilinear models over exponential ones.

BACKGROUND: There is considerable controversy concerning the exact growth profile of size parameters during the cell cycle. Linear, exponential and bilinear models are commonly considered, and the same model may not apply for all species. Selection of the most adequate model to describe a given data-set requires the use of quantitative model selection criteria, such as the partial (sequential) F-test, the Akaike information criterion and the Schwarz Bayesian information criterion, which are suitable for comparing differently parameterized models in terms of the quality and robustness of the fit but have not yet been used in cell growth-profile studies. RESULTS: Length increase data from representative individual fission yeast (Schizosaccharomyces pombe) cells measured on time-lapse films have been reanalyzed using these model selection criteria. To fit the data, an extended version of a recently introduced linearized biexponential (LinBiExp) model was developed, which makes possible a smooth, continuously differentiable transition between two linear segments and, hence, allows fully parametrized bilinear fittings. Despite relatively small differences, essentially all the quantitative selection criteria considered here indicated that the bilinear model was somewhat more adequate than the exponential model for fitting these fission yeast data. CONCLUSION: A general quantitative framework was introduced to judge the adequacy of bilinear versus exponential models in the description of growth time-profiles. For single cell growth, because of the relatively limited data-range, the statistical evidence is not strong enough to favor one model clearly over the other and to settle the bilinear versus exponential dispute. Nevertheless, for the present individual cell growth data for fission yeast, the bilinear model seems more adequate according to all metrics, especially in the case of wee1Delta cells.

Biophysics↗
PubMed2000

The generic viewpoint assumption and Bayesian inference.

The task of human vision is to reliably infer useful information about the external environment from images formed on the retinae. In general, the inference of scene properties from retinal images is not deductive; it requires knowledge about the external environment. Further, it has been suggested that the environment must be regular in some way in order for any scene properties to be reliably inferred. In particular, Knill and Kersten [1991, in Pattern Recognition by Man and Machine Ed. R J Watt (London: Macmillan)] and Jepson et al [1996, in Bayesian Approaches to Perception Eds D Knill, W Richards (Cambridge: Cambridge University Press)] claim that, given an 'unbiased' prior probability distribution for the scenes being observed, the generic viewpoint assumption is not probabilistically valid. However, this claim depends upon the use of representation spaces that may not be appropriate for the problems they consider. In fact, it is problematic to define a rigorous criterion for a probability distribution to be considered 'random' or 'regularity-free' in many natural domains of interest. This problem is closely related to Bertrand's paradox. I propose that, in the case of 'unbiased' priors, the reliability of inferences based on the generic viewpoint assumption depends partly on whether or not an observed coincidence in the image involves features known to be on the same object. This proposal is based on important differences between the distributions associated with: (i) a 'random' placement of features in 3-D, and (ii) the positions of features on a 'randomly shaped' and 'randomly posed' 3-D object. Similar considerations arise in the case of inferring 3-D motion from image motion.

Bayes Theorem↗
PubMed2004

Trading variance reduction with unbiasedness: the regularized subspace information criterion for robust model selection in kernel regression.

A well-known result by Stein (1956) shows that in particular situations, biased estimators can yield better parameter estimates than their generally preferred unbiased counterparts. This letter follows the same spirit, as we will stabilize the unbiased generalization error estimates by regularization and finally obtain more robust model selection criteria for learning. We trade a small bias against a larger variance reduction, which has the beneficial effect of being more precise on a single training set. We focus on the subspace information criterion (SIC), which is an unbiased estimator of the expected generalization error measured by the reproducing kernel Hilbert space norm. SIC can be applied to the kernel regression, and it was shown in earlier experiments that a small regularization of SIC has a stabilization effect. However, it remained open how to appropriately determine the degree of regularization in SIC. In this article, we derive an unbiased estimator of the expected squared error, between SIC and the expected generalization error and propose determining the degree of regularization of SIC such that the estimator of the expected squared error is minimized. Computer simulations with artificial and real data sets illustrate that the proposed method works effectively for improving the precision of SIC, especially in the high-noise-level cases. We furthermore compare the proposed method to the original SIC, the cross-validation, and an empirical Bayesian method in ridge parameter selection, with good results.

Bayes Theorem↗
PubMed2005

Multiple solar particle event dose time profile predictions using Bayesian inference.

The prediction of solar particle event occurrence and the resulting effects on humans and electronics continues to be a mission and/or life-threatening concern for the National Aeronautics and Space Administration and military and commercial satellite operators. While the frequency of events generally follows the solar cycle, individual event occurrence is sporadic and the prediction of resulting effects prior to the event onset is difficult. In one approach to space weather prediction, the forecaster begins to make predictions after the onset of an event. Previous work proved the efficacy of a forecasting methodology that used Bayesian inference and dose and/or dose rate information obtained early after the onset of an event to make predictions of dose and dose rate time profiles out to 120 h beyond onset. The previous work, however, was restricted to predictions for single-event solar particle events. Some of the largest recorded events, including the October 1989 and August 1972 events, were actually multiple events. In this study, we present an analysis of nine large events, some single and some multiple. This work ties together particle flux and fluence data with dose rate and dose calculations in an effort to develop a criterion for characterising an event as multiple and thus, generalising the Bayesian methodology to allow predictions for all events. Dose time profile predictions are made for the four separate events that made up the October 1989 event.

Bayes Theorem↗
PubMed2001

Projections of lung cancer mortality in West Germany: a case study in Bayesian prediction.

We apply a generalized Bayesian age-period-cohort (APC) model to a data-set on lung cancer mortality in West Germany, in the period 1952-1996. Our goal is to predict future death rates until the year 2010, separately for males and females. Since age and period are not measured on the same grid, we propose a generalized APC model where consecutive cohort parameters represent strongly overlapping birth cohorts. This approach results in a rather large number of parameters, where standard algorithms for statistical inference by Markov chain Monte Carlo methods turn out to be computationally intensive. We propose a more efficient implementation based on ideas of block sampling from the time series literature. We entertain two different formulations, penalizing either first or second differences of age, period and cohort parameters. To assess the predictive quality of both formulations, we first forecast the rates for the period 1987-1996 based on data until 1986. A comparison with the actual observed rates is made based on a predictive deviance criterion. Predictions of lung cancer mortality until 2010 are then reported and a modification of the formulation in order to include information on cigarette consumption is finally described.To whom correspondence should be addressed. Currently at Imperial College School of Medicine, Department of Epidemiology and Public Health, Norfolk Place, London W2 1PG, UK.

Journal Article↗
PubMed2006

Fuzzy Markov random fields versus chains for multispectral image segmentation.

This paper deals with a comparison of recent statistical models based on fuzzy Markov random fields and chains for multispectral image segmentation. The fuzzy scheme takes into account discrete and continuous classes which model the imprecision of the hidden data. In this framework, we assume the dependence between bands and we express the general model for the covariance matrix. A fuzzy Markov chain model is developed in an unsupervised way. This method is compared with the fuzzy Markovian field model previously proposed by one of the authors. The segmentation task is processed with Bayesian tools, such as the well-known MPM (Mode of Posterior Marginals) criterion. Our goal is to compare the robustness and rapidity for both methods (fuzzy Markov fields versus fuzzy Markov chains). Indeed, such fuzzy-based procedures seem to be a good answer, e.g., for astronomical observations when the patterns present diffuse structures. Moreover, these approaches allow us to process missing data in one or several spectral bands which correspond to specific situations in astronomy. To validate both models, we perform and compare the segmentation on synthetic images and raw multispectral astronomical data.

Algorithms↗
PubMed2005

MEG source localization under multiple constraints: an extended Bayesian framework.

To use Electroencephalography (EEG) and Magnetoencephalography (MEG) as functional brain 3D imaging techniques, identifiable distributed source models are required. The reconstruction of EEG/MEG sources rests on inverting these models and is ill-posed because the solution does not depend continuously on the data and there is no unique solution in the absence of prior information or constraints. We have described a general framework that can account for several priors in a common inverse solution. An empirical Bayesian framework based on hierarchical linear models was proposed for the analysis of functional neuroimaging data [Friston, K., Penny, W., Phillips, C., Kiebel, S., Hinton, G., Ashburner, J., 2002. Classical and Bayesian inference in neuroimaging: theory. NeuroImage 16, 465-483] and was evaluated recently in the context of EEG [Phillips, C., Mattout, J., Rugg, M.D., Maquet, P., Friston, K., 2005. An empirical Bayesian solution to the source reconstruction problem in EEG. NeuroImage 24, 997-1011]. The approach consists of estimating the expected source distribution and its conditional variance that is constrained by an empirically determined mixture of prior variance components. Estimation uses Expectation-Maximization (EM) to give the Restricted Maximum Likelihood (ReML) estimate of the variance components (in terms of hyperparameters) and the Maximum A Posteriori (MAP) estimate of the source parameters. In this paper, we extend the framework to compare different combinations of priors, using a second level of inference based on Bayesian model selection. Using Monte-Carlo simulations, ReML is first compared to a classic Weighted Minimum Norm (WMN) solution under a single constraint. Then, the ReML estimates are evaluated using various combinations of priors. Both standard criterion and ROC-based measures were used to assess localization and detection performance. The empirical Bayes approach proved useful as: (1) ReML was significantly better than WMN for single priors; (2) valid location priors improved ReML source localization; (3) invalid location priors did not significantly impair performance. Finally, we show how model selection, using the log-evidence, can be used to select the best combination of priors. This enables a global strategy for multiple prior-based regularization of the MEG/EEG source reconstruction.

Bayes Theorem↗
PubMed1995

Bayesian design criteria: computation, comparison, and application to a pharmacokinetic and a pharmacodynamic model.

In this paper 3 criteria to design experiments for Bayesian estimation of the parameters of nonlinear models with respect to their parameters, when a prior distribution is available, are presented: the determinant of the Bayesian information matrix, the determinant of the pre-posterior covariance matrix, and the expected information provided by an experiment. A procedure to simplify the computation of these criteria is proposed in the case of continuous prior distributions and is compared with the criterion obtained from a linearization of the model about the mean of the prior distribution for the parameters. This procedure is applied to two models commonly encountered in the area of pharmacokinetics and pharmacodynamics: the one-compartment open model with bolus intravenous single-dose injection and the Emax model. They both involve two parameters. Additive as well as multiplicative gaussian measurement errors are considered with normal prior distributions. Various combinations of the variances of the prior distribution and of the measurement error are studied. Our attention is restricted to designs with limited numbers of measurements (1 or 2 measurements). This situation often occurs in practice when Bayesian estimation is performed. The optimal Bayesian designs that result vary with the variances of the parameter distribution and with the measurement error. The two-point optimal designs sometimes differ from the D-optimal designs for the mean of the prior distribution and may consist of replicating measurements. For the studied cases, the determinant of the Bayesian information matrix and its linearized form lead to the same optimal designs. In some cases, the pre-posterior covariance matrix can be far from its lower bound, namely, the inverse of the Bayesian information matrix, especially for the Emax model and a multiplicative measurement error. The expected information provided by the experiment and the determinant of the pre-posterior covariance matrix generally lead to the same designs except for the Emax model and the multiplicative measurement error. Results show that these criteria can be easily computed and that they could be incorporated in modules for designing experiments.

Bayes Theorem↗
PubMed2005

Relevance feedback using generalized Bayesian framework with region-based optimization learning.

This paper presents a generalized Bayesian framework for relevance feedback in content-based image retrieval. The proposed feedback technique is based on the Bayesian learning method and incorporates a time-varying user model into the formulation. We define the user model with two terms: a target query and a user conception. The target query is aimed to learn the common features from relevant images so as to specify the user's ideal query. The user conception is aimed to learn a parameter set to determine the time-varying matching criterion. Therefore, at each feedback step, the learning process updates not only the target distribution, but also the target query and the matching criterion. In addition, another objective of this paper is to conduct the relevance feedback on images represented in region level. We formulate the matching criterion using a weighting scheme and proposed a region clustering technique to determine the region correspondence between relevant images. With the proposed region clustering technique, we derive a representation in region level to characterize the target query. Experiments demonstrate that the proposed method combined with time-varying user model indeed achieves satisfactory results and our proposed region-based techniques further improve the retrieval accuracy.

Algorithms↗
PubMed2003

Molecular systematics and adaptive radiation of Hawaii's endemic Damselfly genus Megalagrion (Odonata: Coenagrionidae).

Damselflies of the endemic Hawaiian genus Megalagrion have radiated into a wide variety of habitats and are an excellent model group for the study of adaptive radiation. Past phylogenetic analysis based on morphological characters has been problematic. Here, we examine relationships among 56 individuals from 20 of the 23 described species using maximum likelihood (ML) and Bayesian phylogenetic analysis of mitochondrial (1287 bp) and nuclear (1039 bp) DNA sequence data. Models of evolution were chosen using the Akaike information criterion. Problems with distant outgroups were accommodated by constraining the best ML ingroup topology but allowing the outgroups to attach to any ingroup branch in a bootstrap analysis. No strong contradictions were obtained between either data partition and the combined data set. Areas of disagreement are mainly confined to clades that are strongly supported by the mitochondrial DNA and weakly supported by the elongation factor 1alpha data because of lack of changes. However, the combined analysis resulted in a unique tree. Correlation between Bayesian posterior probabilities and bootstrap percentages decreased in concert with decreasing information in the data partitions. In cases where nodes were supported by single characters bootstrap proportions were dramatically reduced compared with posterior probabilities. Two speciation patterns were evident from the phylogenetic analysis. First, most speciation is interisland and occurred as members of established ecological guilds colonized new volcanoes after they emerged from the sea. Second, there are several instances of rapid radiation into a variety of specialized habitats, in one case entirely within the island of Kauai. Application of a local clock procedure to the mitochondrial DNA topology suggests that two of these radiations correspond to the development of habitat on the islands of Kauai and Oahu. About 4.0 million years ago, species simultaneously moved into fast streams and plant leaf axils on Kauai, and about 1.5 million years later another group moved simultaneously to seeps and terrestrial habitats on Oahu. Results from the local clock analysis also strongly suggest that Megalagrion arrived in Hawaii about 10 million years ago, well before the emergence of Kauai. Date estimates were more sensitive to the particular node that was fixed in time than to the model of local branch evolution used. We propose a general model for the development of endemic damselfly species on Hawaiian Islands and document five potential cases of hybridization (M. xanthomelas x M. pacificum, M. eudytum x M. vagabundum, M. orobates x M. oresitrophum, M. nesiotes x M. oahuense, and M. mauka x M. paludicola).

Adaptation, Biological↗