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PubMed1995

A Bayesian group sequential design for a multiple arm randomized clinical trial.

Group sequential designs for randomized clinical trials allow analyses of accruing data. Most group sequential designs in the literature concern the comparison of two treatments and maintain an overall prespecified type I error. As the number of treatments increases, however, so does the probability of falsely rejecting the null hypothesis. Bayesian statisticians concern themselves with the observed data and abide by the likelihood principle. As long as previous analyses do not change the likelihood, these analyses do not change Bayesian inference. In this paper, we discuss a group sequential design for a proposed randomized clinical trial comparing four treatment regimens. Bayesian ideas underlie the design and posterior probability calculations determine the criteria for stopping accrual to one or more of the treatments. We use computer simulation to estimate the frequentists properties of the design, information of interest to many of our collaborators. We show that relatively simple posterior probability calculations, along with simulations to calculate power under alternative hypotheses, can produce appealing designs for randomized clinical trials.

Bayes Theorem↗
PubMed2005

Bayesian estimation of diagnostic tests accuracy for semi-latent data with covariates.

The performance of a diagnostic test is usually summarized by its sensitivity and specificity. Sensitivity is the probability of a positive result, once the individual is truly ill, and specificity is the probability of a negative result, regarding a healthy individual. These measures are obtained by comparing the test outcome and the results of a reference test generically denominated gold standard. However, in many applied problems considering two diagnostic tests, the gold standard is not available for those individuals with negative results on both tests. In addition, not all diagnostic tests have the same performance across different populations. In this context, we present a Bayesian inference approach for performance measures estimation and we propose an extension of this procedure involving the inclusion of covariates. This Bayesian approach is based on Markov Chain Monte Carlo methods. The conditional dependence between the diagnostic tests was considered. As an example, we applied the proposed methodology to a real data set obtained from the medical literature.

Bayes Theorem↗
PubMed2005

Phylogeny of Tubificidae (Annelida, Clitellata) based on mitochondrial and nuclear sequence data.

The tubificid clitellates are a common component in the freshwater bottom fauna and are also the most abundant oligochaete group in marine habitats. There are over 800 described species classified in six subfamilies; Tubificinae, Limnodriloidinae, Rhyacodrilinae, Telmatodrilinae, Phallodrilinae, and Naidinae. In this study we examine the phylogenetic relationships in Tubificidae using a combination of mitochondrial 16S rDNA and nuclear 18S rDNA sequence data. Sequences were obtained from five outgroup and 56 ingroup taxa, including five of the six subfamilies of Tubificidae. The data were analysed by maximum parsimony and Bayesian inference. The resulting tree topologies are virtually without conflict. Several associations traditionally recognized within the family Tubificidae are supported, in the Bayesian analysis including a sister group relationship between Tubificinae and Limnodriloidinae. The results also indicate that Rhyacodrilinae is polyphyletic--some of its members (Heterodrilus spp.) fall into a clade with Phallodrilinae, all other groups with Naidinae. Naidinae is also polyphyletic with two rhyacodriline genera, Monopylephorus and Ainudrilus, nested within. Most of the tubificid genera included in the study are supported as monophyletic; however, Tubifex and Limnodriloides are refuted, and Tubificoides is unresolved from other tubificine taxa.

Animals↗
PubMed2003

Comparison of Bayesian and maximum likelihood bootstrap measures of phylogenetic reliability.

Owing to the exponential growth of genome databases, phylogenetic trees are now widely used to test a variety of evolutionary hypotheses. Nevertheless, computation time burden limits the application of methods such as maximum likelihood nonparametric bootstrap to assess reliability of evolutionary trees. As an alternative, the much faster Bayesian inference of phylogeny, which expresses branch support as posterior probabilities, has been introduced. However, marked discrepancies exist between nonparametric bootstrap proportions and Bayesian posterior probabilities, leading to difficulties in the interpretation of sometimes strongly conflicting results. As an attempt to reconcile these two indices of node reliability, we apply the nonparametric bootstrap resampling procedure to the Bayesian approach. The correlation between posterior probabilities, bootstrap maximum likelihood percentages, and bootstrapped posterior probabilities was studied for eight highly diverse empirical data sets and were also investigated using experimental simulation. Our results show that the relation between posterior probabilities and bootstrapped maximum likelihood percentages is highly variable but that very strong correlations always exist when Bayesian node support is estimated on bootstrapped character matrices. Moreover, simulations corroborate empirical observations in suggesting that, being more conservative, the bootstrap approach might be less prone to strongly supporting a false phylogenetic hypothesis. Thus, apparent conflicts in topology recovered by the Bayesian approach were reduced after bootstrapping. Both posterior probabilities and bootstrap supports are of great interest to phylogeny as potential upper and lower bounds of node reliability, but they are surely not interchangeable and cannot be directly compared.

Animals↗
PubMedDate not supplied

Bayesian approaches to meta-analysis of ROC curves.

A comparative review of important classic and Bayesian approaches to fixed-effects and random-effects meta-analysis of binormal ROC curves and areas underneath them is presented. The ROC analyses results of seven evaluation studies concerning the dexamethasone suppression test provide the basis for a worked example. Particular attention is given to fully Bayesian inference, a novelty in the ROC context, based on Gibbs samples from posterior distributions of hierarchical model parameters and related quantities. Fully Bayesian meta-analysis may properly account for the uncertainty associated with the model parameters, possibly incorporating prior knowledge and beliefs, and allows clinically intuitive predictions of unobserved study effects via calculation of posterior predictive densities. The effects of various different prior specifications (six noninformative as well as one informative) on the posterior estimates are investigated (sensitivity-analysis). Recommendations and suggestions for further research are made. Computer code for the more advanced methods may either be downloaded via the Internet or be found elsewhere.

Bayes Theorem↗
PubMed2006

Prior specification in Bayesian statistics: three cautionary tales.

One of the most important differences between Bayesian and traditional techniques is that the former combines information available beforehand-captured in the prior distribution and reflecting the subjective state of belief before an experiment is carried out-and what the data teach us, as expressed in the likelihood function. Bayesian inference is based on the combination of prior and current information which is reflected in the posterior distribution. The fast growing implementation of Bayesian analysis techniques can be attributed to the development of fast computers and the availability of easy to use software. It has long been established that the specification of prior distributions should receive a lot of attention. Unfortunately, flat distributions are often (inappropriately) used in an automatic fashion in a wide range of types of models. We reiterate that the specification of the prior distribution should be done with great care and support this through three examples. Even in the absence of strong prior information, prior specification should be done at the appropriate scale of biological interest. This often requires incorporation of (weak) prior information based on common biological sense. Very weak and uninformative priors at one scale of the model may result in relatively strong priors at other levels affecting the posterior distribution. We present three different examples intuïvely illustrating this phenomenon indicating that this bias can be substantial (especially in small samples) and is widely present. We argue that complete ignorance or absence of prior information may not exist. Because the central theme of the Bayesian paradigm is to combine prior information with current data, authors should be encouraged to publish their raw data such that every scientist is able to perform an analysis incorporating his/her own (subjective) prior distributions.

Animals↗
PubMed2000

Accommodating phylogenetic uncertainty in evolutionary studies.

Many evolutionary studies use comparisons across species to detect evidence of natural selection and to examine the rate of character evolution. Statistical analyses in these studies are usually performed by means of a species phylogeny to accommodate the effects of shared evolutionary history. The phylogeny is usually treated as known without error; this assumption is problematic because inferred phylogenies are subject to both stochastic and systematic errors. We describe methods for accommodating phylogenetic uncertainty in evolutionary studies by means of Bayesian inference. The methods are computationally intensive but general enough to be applied in most comparative evolutionary studies.

Animals↗
PubMed2005

Phylogenetic relationships and evolutionary traits in Ranunculus s.l. (Ranunculaceae) inferred from ITS sequence analysis.

Ranunculus is a large genus with a worldwide distribution. Phylogenetic analyses of c. 200 species of Ranunculus s.l. based on sequences of the nrITS using maximum parsimony and Bayesian inference yielded high congruence with previous cpDNA restriction site analyses, but strongly contradict previous classifications. A large core clade including Ranunculus subg. Ranunculus, subg. Batrachium, subg. Crymodes p.p., Ceratocephala, Myosurus, and Aphanostemma is separated from R. subg. Ficaria, subg. Pallasiantha, subg. Coptidium, subg. Crymodes p.p., Halerpestes, Peltocalathos, Callianthemoides, and Arcteranthis. Within the core clade, 19 clades can be described with morphological and karyological features. Several sections are not monophyletic. Parallel evolution of morphological characters in adaptation to climatic conditions may be a reason for incongruence of molecular data and morphology-based classifications. In some mountainous regions, groups of closely related species may have originated from adaptive radiation and rapid speciation. Split decomposition analysis indicated complex patterns of relationship and suggested hybridization in the apomictic R. auricomus complex, R. subg. Batrachium, and the white-flowering European alpines. The evolutionary success of the genus might be due to a combination of morphological plasticity and adaptations, hybridization and polyploidy as important factors for regional diversification, and a broad range of reproductive strategies.

DNA, Plant↗
PubMed2006

Molecular phylogeny of the Branchiopoda (Crustacea)--multiple approaches suggest a 'diplostracan' ancestry of the Notostraca.

The evolutionary history of Branchiopoda (Crustacea) traditionally has attracted considerable interest due to the diversity of the group. Recently molecular methods have been applied to the study of branchiopod systematics with some success, but central questions, such as the phylogenetic position of Laevicaudata and Notostraca, and the intrinsic cladoceran phylogeny, remain unanswered. We examined the phylogeny of Branchiopoda by using two genes, mitochondrial 16S rRNA and nuclear 28S rRNA, which previously have seen little use for inferring branchiopod phylogeny. The number of ingroup taxa included was 42, representing all eight extant branchiopod orders. The data were analyzed using parsimony, maximum likelihood, and Bayesian Inference of phylogeny. Some higher-level taxa were supported in all analyses of the combined data: Phyllopoda, Cladoceromorpha, Cladocera, and Gymnomera. Other higher-level taxa were not supported in any trees: Diplostraca and Conchostraca. A case is made for a possible diplostracan ingroup position of Notostraca based on our data and on previously published molecular and morphological evidence. The recent discovery of a Devonian branchiopod, which is morphologically an intermediate between a notostracan and a 'conchostracan', is congruent with a diplostracan ancestry of Notostraca.

Animals↗
PubMed1998

Bayesian protein family classifier.

A Bayesian procedure for the simultaneous alignment and classification of sequences into subclasses is described. This Gibbs sampling algorithm iterates between an alignment step and a classification step. It employs Bayesian inference for the identification of the number of conserved columns, the number of motifs in each class, their size, and the size of the classes. Using Bayesian prediction, inter-class differences in all these variables are brought to bare on the classification. Application to a superfamily of cyclic nucleotide-binding proteins identifies both similarities and differences in the sequence characteristics of the five subclasses identified by the procedure: 1) cNMP-dependent kinases, 2) prokaryotic cAMP-dependent regulatory proteins, CRP-type, 3) prokaryotic regulatory proteins, FNR-type, 4) cAMP gated ion channel proteins of animals, and 5) cAMP gated ion channels of plants.

Algorithms↗
PubMed2001

Directed Convergence in Stable Percept Acquisition.

We view a perceptual capacity as a nondeductive inference, represented as a function from a set of premises to a set of conclusions. The application of the function to a single premise to produce a single conclusion is called a "percept" or "instantaneous percept." We define a stable percept as a convergent sequence of instantaneous percepts. Assuming that the sets of premises and conclusions are metric spaces, we introduce a strategy for acquiring stable percepts, called directed convergence. We consider probabilistic inferences, where the premise and conclusion sets are spaces of probability measures, and in this context we study Bayesian probabilistic/recursive inference. In this type of Bayesian inference the premises are probability measures, and the prior as well as the posterior is updated nontrivially at each iteration. This type of Bayesian inference is distinguished from classical Bayesian statistical inference where the prior remains fixed, and the posterior evolves by conditioning on successively more punctual premises. We indicate how the directed convergence procedure may be implemented in the context of Bayesian probabilistic/recursive inference. We discuss how the L(infinity) metric can be used to give numerical control of this type of Bayesian directed convergence. Copyright 2001 Academic Press.

Journal Article↗
PubMed1993

An introduction to the Bayesian analysis of clinical trials.

Although most clinical trials comparing therapies are analyzed using classical hypothesis testing and P values, such methods do not yield the information most useful to the clinician, that is, the probability that one treatment is more efficacious than another. Bayesian inference can yield this probability but only if we quantify our prior beliefs about the possible efficacies of the treatments studied. This article gives a brief introduction to Bayesian methods and contrasts them with classical hypothesis testing. It shows that the quantification of prior beliefs is a common and necessary part of the interpretation of clinical information, whether from a laboratory test or published clinical trial. Advantages of Bayesian analysis over classical analysis of clinical trials include the ability to incorporate prior information regarding treatment efficacies into the analysis; the ability to make multiple unscheduled inspections of accumulating data without increasing the error rate of the study; and the ability to calculate the probability that one treatment is more effective than another. Because it is likely that Bayesian methods will be used more often in the analysis of future clinical trials, investigators and readers should be aware of the two schools of statistical thought and the strengths and weaknesses of each.

Bayes Theorem↗
PubMed2001

Inference for an epidemic when susceptibility varies.

A stochastic epidemic model featuring fixed-length latent periods, gamma-distributed infectious periods and randomly varying heterogeneity among susceptibles is considered. A Markov chain Monte Carlo algorithm is developed for performing Bayesian inference for the parameters governing the infectious-period length and the hyper-parameters governing the heterogeneity of susceptibility. This method of analysis applies to a wider class of diseases than methods proposed previously. An application to smallpox data confirms results about heterogeneity suggested by an earlier analysis that relied on less realistic assumptions.

Journal Article↗
PubMed2003

Bayesian hierarchical approach to estimate insulin sensitivity by minimal model.

We adopted Bayesian analysis in combination with hierarchical (population) modelling to estimate simultaneously population and individual insulin sensitivity (SI) and glucose effectiveness (SG) with the minimal model of glucose kinetics using data collected during insulin-modified intravenous glucose tolerance test (IVGTT) and made comparison with the standard non-linear regression analysis. After fasting overnight, subjects with newly presenting Type II diabetes according to World Health Organization criteria (n =65; 53 males, 12 females; age, 54 +/- 9 years; body mass index, 30.4 +/- 5.2 kg/m2; means+/-S.D.) underwent IVGTT consisting of a 0.3 g of glucose bolus/kg of body weight given at time zero for 2 min, followed by 0.05 unit of insulin/kg of body weight at 20 min. Bayesian inference was carried out using vague prior distributions and log-normal distributions to guarantee non-negativity and, thus, physiological plausibility of model parameters and associated credible intervals. Bayesian analysis gave estimates of SI in all subjects. Non-linear regression analysis failed in four cases, where Bayesian analysis-derived SI was located in the lower quartile and was estimated with lower precision. The population means of SI and SG provided by Bayesian analysis and non-linear regression were identical, but the interquartile range given by Bayesian analysis was tighter by approx. 20% for SI and by approx. 15% for SG. Individual insulin sensitivities estimated by the two methods were highly correlated ( rS=0.98; P <0.001). However, the correlation in the lower 20% centile of the insulin-sensitivity range was significantly lower than the correlation in the upper 80% centile ( rS=0.71 compared with rS=0.99; P <0.001). We conclude that the Bayesian hierarchical analysis is an appealing method to estimate SI and SG, as it avoids parameter estimation failures, and should be considered when investigating insulin-resistant subjects.

Adult↗
PubMed2003

Intelligent machines in the twenty-first century: foundations of inference and inquiry.

The last century saw the application of Boolean algebra to the construction of computing machines, which work by applying logical transformations to information contained in their memory. The development of information theory and the generalization of Boolean algebra to Bayesian inference have enabled these computing machines, in the last quarter of the twentieth century, to be endowed with the ability to learn by making inferences from data. This revolution is just beginning as new computational techniques continue to make difficult problems more accessible. Recent advances in our understanding of the foundations of probability theory have revealed implications for areas other than logic. Of relevance to intelligent machines, we recently identified the algebra of questions as the free distributive algebra, which will now allow us to work with questions in a way analogous to that which Boolean algebra enables us to work with logical statements. In this paper, we examine the foundations of inference and inquiry. We begin with a history of inferential reasoning, highlighting key concepts that have led to the automation of inference in modern machine-learning systems. We then discuss the foundations of inference in more detail using a modern viewpoint that relies on the mathematics of partially ordered sets and the scaffolding of lattice theory. This new viewpoint allows us to develop the logic of inquiry and introduce a measure describing the relevance of a proposed question to an unresolved issue. Last, we will demonstrate the automation of inference, and discuss how this new logic of inquiry will enable intelligent machines to ask questions. Automation of both inference and inquiry promises to allow robots to perform science in the far reaches of our solar system and in other star systems by enabling them not only to make inferences from data, but also to decide which question to ask, which experiment to perform, or which measurement to take given what they have learned and what they are designed to understand.

Artificial Intelligence↗
PubMed2002

Bayesian framework for least-squares support vector machine classifiers, gaussian processes, and kernel Fisher discriminant analysis.

The Bayesian evidence framework has been successfully applied to the design of multilayer perceptrons (MLPs) in the work of MacKay. Nevertheless, the training of MLPs suffers from drawbacks like the nonconvex optimization problem and the choice of the number of hidden units. In support vector machines (SVMs) for classification, as introduced by Vapnik, a nonlinear decision boundary is obtained by mapping the input vector first in a nonlinear way to a high-dimensional kernel-induced feature space in which a linear large margin classifier is constructed. Practical expressions are formulated in the dual space in terms of the related kernel function, and the solution follows from a (convex) quadratic programming (QP) problem. In least-squares SVMs (LS-SVMs), the SVM problem formulation is modified by introducing a least-squares cost function and equality instead of inequality constraints, and the solution follows from a linear system in the dual space. Implicitly, the least-squares formulation corresponds to a regression formulation and is also related to kernel Fisher discriminant analysis. The least-squares regression formulation has advantages for deriving analytic expressions in a Bayesian evidence framework, in contrast to the classification formulations used, for example, in gaussian processes (GPs). The LS-SVM formulation has clear primal-dual interpretations, and without the bias term, one explicitly constructs a model that yields the same expressions as have been obtained with GPs for regression. In this article, the Bayesian evidence framework is combined with the LS-SVM classifier formulation. Starting from the feature space formulation, analytic expressions are obtained in the dual space on the different levels of Bayesian inference, while posterior class probabilities are obtained by marginalizing over the model parameters. Empirical results obtained on 10 public domain data sets show that the LS-SVM classifier designed within the Bayesian evidence framework consistently yields good generalization performances.

Artificial Intelligence↗
PubMed2000

Bayesian estimation of range for microsatellite loci.

Microsatellite loci have become important in population genetics because of their high level of polymorphism in natural populations, very frequent occurrence throughout the genome, and apparently high mutation rate. Observed repeat numbers (alleles size) in natural populations and expectations based on computer simulations suggest that the range of repeat numbers at a microsatellite locus is restricted. This range is a key parameter that should be properly estimated in order to proceed with calculations of divergence times in phylogenetic studies and to better investigate the within- and between-population variability. The 'plug-in' estimate of range based on the minimum and maximum value observed in a sample is not satisfactory because of the relatively large number of alleles in comparison with typical sample sizes. In this paper, a set of data from 30 dinucleotide microsatellite loci is analysed under the assumption of independence among loci. Bayesian inference on range for one locus is obtained by assuming that constraints on range values exist as sharp bounds. Closed-form calculations and robustness revealed by our analysis suggest that the proposed Bayesian approach might be routinely used by researchers to classify microsatellite loci according to the estimated value of their allelic range.

Bayes Theorem↗
PubMed2006

Evaluation of decay times in coupled spaces: an efficient search algorithm within the Bayesian framework.

This paper discusses an efficient method for evaluating multiple decay times within the Bayesian framework. Previous works [N. Xiang and P. M. Goggans, J. Acoust. Soc. Am. 110, 1415-1424 (2001); 113, 2685-2697 (2003); N. Xiang, P. M. Goggans, T. Jasa, and M. Kleiner, 117, 3707-3715 (2005)] have applied the Bayesian inference to cope with demanding tasks in estimating multiple decay times from Schroeder decay functions measured or calculated in acoustically coupled spaces. Since then a number of recent works call for efficient estimation methods within the Bayesian framework. An efficient analysis is of practical significance for better understanding and modeling the sound energy decay process in acoustically coupled spaces or even in single spaces for reverberation time estimation. This paper will first formulate the Bayesian posterior probability distribution function (PPDF) in a matrix form to reduce the dimensionality as applied to the decay time evaluation. Based on existence of only global extremes of PPDFs as observed from extensive experimental data, this paper describes a dedicated search algorithm for an efficient estimation of decay times.

Journal Article↗