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Showing papers on "Bayes' theorem published in 2008"


Journal ArticleDOI
TL;DR: This review is an introduction to Bayesian methods in cosmology and astrophysics and recent results in the field, and presents Bayesian probability theory and its conceptual underpinnings, Bayes' Theorem and the role of priors.
Abstract: The application of Bayesian methods in cosmology and astrophysics has flourished over the past decade, spurred by data sets of increasing size and complexity. In many respects, Bayesian methods have proven to be vastly superior to more traditional statistical tools, offering the advantage of higher efficiency and of a consistent conceptual basis for dealing with the problem of induction in the presence of uncertainty. This trend is likely to continue in the future, when the way we collect, manipulate and analyse observations and compare them with theoretical models will assume an even more central role in cosmology. This review is an introduction to Bayesian methods in cosmology and astrophysics and recent results in the field. I first present Bayesian probability theory and its conceptual underpinnings, Bayes' Theorem and the role of priors. I discuss the problem of parameter inference and its general solution, along with numerical techniques such as Monte Carlo Markov Chain methods. I then review the th...

962 citations


Journal ArticleDOI
TL;DR: The Bayesian modelling methods introduced in this article represent an array of enhanced tools for learning the genetic structure of populations designed to meet the increasing need for analyzing large-scale population genetics data.
Abstract: During the most recent decade many Bayesian statistical models and software for answering questions related to the genetic structure underlying population samples have appeared in the scientific literature. Most of these methods utilize molecular markers for the inferences, while some are also capable of handling DNA sequence data. In a number of earlier works, we have introduced an array of statistical methods for population genetic inference that are implemented in the software BAPS. However, the complexity of biological problems related to genetic structure analysis keeps increasing such that in many cases the current methods may provide either inappropriate or insufficient solutions. We discuss the necessity of enhancing the statistical approaches to face the challenges posed by the ever-increasing amounts of molecular data generated by scientists over a wide range of research areas and introduce an array of new statistical tools implemented in the most recent version of BAPS. With these methods it is possible, e.g., to fit genetic mixture models using user-specified numbers of clusters and to estimate levels of admixture under a genetic linkage model. Also, alleles representing a different ancestry compared to the average observed genomic positions can be tracked for the sampled individuals, and a priori specified hypotheses about genetic population structure can be directly compared using Bayes' theorem. In general, we have improved further the computational characteristics of the algorithms behind the methods implemented in BAPS facilitating the analyses of large and complex datasets. In particular, analysis of a single dataset can now be spread over multiple computers using a script interface to the software. The Bayesian modelling methods introduced in this article represent an array of enhanced tools for learning the genetic structure of populations. Their implementations in the BAPS software are designed to meet the increasing need for analyzing large-scale population genetics data. The software is freely downloadable for Windows, Linux and Mac OS X systems at http://web.abo.fi/fak/mnf//mate/jc/software/baps.html .

818 citations


Journal ArticleDOI
TL;DR: It was from here that "Bayesian" ideas first spread through the mathematical world, as Bayes's own article was ignored until 1780 and played no important role in scientific debate until the 20th century.
Abstract: The influence of this Thomas Bayes' work was immense. It was from here that "Bayesian" ideas first spread through the mathematical world, as Bayes's own article was ignored until 1780 and played no important role in scientific debate until the 20th century. It was also this article of Laplace's that introduced the mathematical techniques for the asymptotic analysis of posterior distributions that are still employed today. And it was here that the earliest example of optimum estimation can be found, the derivation and characterization of an estimator that minimized a particular measure of posterior expected loss. After more than two centuries, we mathematicians, statisticians cannot only recognize our roots in this masterpiece of our science, we can still learn from it.

774 citations


Book
30 Jun 2008
TL;DR: Approaches for statistical inference Motivating Vignettes Defining the Approaches The Bayes-Frequentist Controversy Some Basic Bayesian Models The Bayesian approach
Abstract: Approaches for statistical inference Introduction Motivating Vignettes Defining the Approaches The Bayes-Frequentist Controversy Some Basic Bayesian Models The Bayes approach Introduction Prior Distributions Bayesian Inference Hierarchical Modeling Model Assessment Nonparametric Methods Bayesian computation Introduction Asymptotic Methods Noniterative Monte Carlo Methods Markov Chain Monte Carlo Methods Model criticism and selection Bayesian Modeling Bayesian Robustness Model Assessment Bayes Factors via Marginal Density Estimation Bayes Factors via Sampling over the Model Space Other Model Selection Methods The empirical Bayes approach Introduction Parametric EB Point Estimation Nonparametric EB Point Estimation Interval Estimation Bayesian Processing and Performance Frequentist Performance Empirical Bayes Performance Bayesian design Principles of Design Bayesian Clinical Trial Design Applications in Drug and Medical Device Trials Special methods and models Estimating Histograms and Ranks Order Restricted Inference Longitudinal Data Models Continuous and Categorical Time Series Survival Analysis and Frailty Models Sequential Analysis Spatial and Spatio-Temporal Models Case studies Analysis of Longitudinal AIDS Data Robust Analysis of Clinical Trials Modeling of Infectious Diseases Appendices Distributional Catalog Decision Theory Answers to Selected Exercises References Author Index Subject Index Index Exercises appear at the end of each chapter.

756 citations


Journal ArticleDOI
TL;DR: The Extended Bayesian Skyline Plot is presented, a non-parametric Bayesian Markov chain Monte Carlo algorithm that extends a previous coalescent-based method in several ways, including the ability to analyze multiple loci, demonstrating the essential role of multiple loco in recovering population size dynamics.
Abstract: Effective population size (N e ) is related to genetic variability and is a basic parameter in many models of population genetics. A number of methods for inferring current and past population sizes from genetic data have been developed since JFC Kingman introduced the n-coalescent in 1982. Here we present the Extended Bayesian Skyline Plot, a non-parametric Bayesian Markov chain Monte Carlo algorithm that extends a previous coalescent-based method in several ways, including the ability to analyze multiple loci. Through extensive simulations we show the accuracy and limitations of inferring population size as a function of the amount of data, including recovering information about evolutionary bottlenecks. We also analyzed two real data sets to demonstrate the behavior of the new method; a single gene Hepatitis C virus data set sampled from Egypt and a 10 locus Drosophila ananassae data set representing 16 different populations. The results demonstrate the essential role of multiple loci in recovering population size dynamics. Multi-locus data from a small number of individuals can precisely recover past bottlenecks in population size which can not be characterized by analysis of a single locus. We also demonstrate that sequence data quality is important because even moderate levels of sequencing errors result in a considerable decrease in estimation accuracy for realistic levels of population genetic variability.

697 citations


Book ChapterDOI
01 Apr 2008
TL;DR: The goal in this work is to illustrate the kinds of computational models of cognition that the authors can build if they assume that human learning and inference approximately follow the principles of Bayesian probabilistic inference.
Abstract: For over 200 years, philosophers and mathematicians have be en using probability theory to describe human cognition. While the theory of prob abilities was first developed as a means of analyzing games of chance, it quickly took on a larger and deeper significance as a formal account of how rational agents should reason in situations of uncertainty (Gigerenzer et al., 1989; Hacking, 1975). Our goal in this ch apter is to illustrate the kinds of computational models of cognition that we can build if we assume that human learning and inference approximately follow the principles of Bayesian probabilistic inference, and to explain some of the mathematical ideas and techniques underlying those models

494 citations


Book ChapterDOI
TL;DR: Bayesian regularized artificial neural networks (BRANNs) as mentioned in this paper are more robust than standard back-propagation nets and can reduce or eliminate the need for lengthy cross-validation.
Abstract: Bayesian regularized artificial neural networks (BRANNs) are more robust than standard back-propagation nets and can reduce or eliminate the need for lengthy cross-validation. Bayesian regularization is a mathematical process that converts a nonlinear regression into a "well-posed" statistical problem in the manner of a ridge regression. The advantage of BRANNs is that the models are robust and the validation process, which scales as O(N2) in normal regression methods, such as back propagation, is unnecessary. These networks provide solutions to a number of problems that arise in QSAR modeling, such as choice of model, robustness of model, choice of validation set, size of validation effort, and optimization of network architecture. They are difficult to overtrain, since evidence procedures provide an objective Bayesian criterion for stopping training. They are also difficult to overfit, because the BRANN calculates and trains on a number of effective network parameters or weights, effectively turning off those that are not relevant. This effective number is usually considerably smaller than the number of weights in a standard fully connected back-propagation neural net. Automatic relevance determination (ARD) of the input variables can be used with BRANNs, and this allows the network to "estimate" the importance of each input. The ARD method ensures that irrelevant or highly correlated indices used in the modeling are neglected as well as showing which are the most important variables for modeling the activity data. This chapter outlines the equations that define the BRANN method plus a flowchart for producing a BRANN-QSAR model. Some results of the use of BRANNs on a number of data sets are illustrated and compared with other linear and nonlinear models.

482 citations


Journal ArticleDOI
TL;DR: In this article, the interplay of Bayesian and frequentist ideas in the two-groups setting, with particular attention focussed on Benjamini and Hochberg's false discovery rate method, is discussed.
Abstract: The classic frequentist theory of hypothesis testing developed by Neyman, Pearson, and Fisher has a claim to being the Twentieth Century’s most influential piece of applied mathematics. Something new is happening in the Twenty-First Century: high throughput devices, such as microarrays, routinely require simultaneous hypothesis tests for thousands of individual cases, not at all what the classical theory had in mind. In these situations empirical Bayes information begins to force itself upon frequentists and Bayesians alike. The two-groups model is a simple Bayesian construction that facilitates empirical Bayes analysis. This article concerns the interplay of Bayesian and frequentist ideas in the two-groups setting, with particular attention focussed on Benjamini and Hochberg’s False Discovery Rate method. Topics include the choice and meaning of the null hypothesis in large-scale testing situations, power considerations, the limitations of permutation methods, significance testing for groups of cases (such as pathways in microarray studies), correlation effects, multiple confidence intervals, and Bayesian competitors to the two-groups model.

396 citations


Journal ArticleDOI
01 Jun 2008-Genetics
TL;DR: Several Bayesian hierarchical models for mapping multiple QTL that simultaneously fit and estimate all possible genetic effects associated with all markers are proposed, including a Bayesian version of the popular least absolute shrinkage and selection operator (LASSO) model and the well-known Student's t model.
Abstract: The mapping of quantitative trait loci (QTL) is to identify molecular markers or genomic loci that influence the variation of complex traits. The problem is complicated by the facts that QTL data usually contain a large number of markers across the entire genome and most of them have little or no effect on the phenotype. In this article, we propose several Bayesian hierarchical models for mapping multiple QTL that simultaneously fit and estimate all possible genetic effects associated with all markers. The proposed models use prior distributions for the genetic effects that are scale mixtures of normal distributions with mean zero and variances distributed to give each effect a high probability of being near zero. We consider two types of priors for the variances, exponential and scaled inverse-χ2 distributions, which result in a Bayesian version of the popular least absolute shrinkage and selection operator (LASSO) model and the well-known Student's t model, respectively. Unlike most applications where fixed values are preset for hyperparameters in the priors, we treat all hyperparameters as unknowns and estimate them along with other parameters. Markov chain Monte Carlo (MCMC) algorithms are developed to simulate the parameters from the posteriors. The methods are illustrated using well-known barley data.

329 citations


01 Jan 2008
TL;DR: The Bayes’ theorem is generalized within the transferable belief model framework and the properties of the DRC and GBT and their uses for belief propagation in directed belief networks are analysed.
Abstract: We generalize the Bayes’ theorem within the transferable belief model framework. The Generalized Bayesian Theorem (GBT) allows us to compute the belief over a space Θ given an observation x⊆ X when one knows only the beliefs over X for every θi ∈ Θ. We also discuss the Disjunctive Rule of Combination (DRC) for distinct pieces of evidence. This rule allows us to compute the belief over X from the beliefs induced by two distinct pieces of evidence when one knows only that one of the pieces of evidence holds. The properties of the DRC and GBT and their uses for belief propagation in directed belief networks are analysed. The use of the discounting factors is justfied. The application of these rules is illustrated by an example of medical diagnosis.

307 citations


Book
28 Feb 2008
TL;DR: In this paper, the Royall Road to Evidence is described as a "road to evidence" and the probability of a hypothesis being true or false is defined as the probability that the hypothesis is true.
Abstract: Karl Popper and Demarcation Kuhn and Lakatos: Paradigms and Programmes Neyman, Pearson and Hypothesis Testing Bayes and the Probability of Hypotheses Fisher and the Likelihood: The Royall Road to Evidence

Journal ArticleDOI
TL;DR: In this article, a simulation framework is developed to generate data from the Poisson-gamma distributions using different values describing the mean, the dispersion parameter, the sample size, and the prior specification.

Journal ArticleDOI
TL;DR: This work implements an enhanced hybrid classification method through the utilization of the naive Bayes approach and the support vector machine (SVM) that shows a significant reduction in training time as compared to the Lsquare method and significant improvement in the classification accuracy when compared to pure naive BayES systems and also the TF-IDF/SVM hybrids.
Abstract: This work implements an enhanced hybrid classification method through the utilization of the naive Bayes approach and the support vector machine (SVM). In this project, the Bayes formula was used to vectorize (as opposed to classify) a document according to a probability distribution reflecting the probable categories that the document may belong to. The Bayes formula gives a range of probabilities to which the document can be assigned according to a predetermined set of topics (categories) such as those found in the "20 Newsgroups" data set for instance. Using this probability distribution as the vectors to represent the document, the SVM can then be used to classify the documents on a multidimensional level. The effects of an inadvertent dimensionality reduction caused by classifying using only the highest probability using the naive Bayes classifier can be overcome using the SVM by employing all the probability values associated with every category for each document. This method can be used for any data set and shows a significant reduction in training time as compared to the Lsquare method and significant improvement in the classification accuracy when compared to pure naive Bayes systems and also the TF-IDF/SVM hybrids.

Journal ArticleDOI
TL;DR: The variance of Bayes factor estimates is investigated, and the stability of the Annealed Importance Sampling and the Annealing-Melting Integration methods are highlighted for the purposes of comparing nonlinear models.
Abstract: Motivation: There often are many alternative models of a biochemical system. Distinguishing models and finding the most suitable ones is an important challenge in Systems Biology, as such model ranking, by experimental evidence, will help to judge the support of the working hypotheses forming each model. Bayes factors are employed as a measure of evidential preference for one model over another. Marginal likelihood is a key component of Bayes factors, however computing the marginal likelihood is a difficult problem, as it involves integration of nonlinear functions in multidimensional space. There are a number of methods available to compute the marginal likelihood approximately. A detailed investigation of such methods is required to find ones that perform appropriately for biochemical modelling. Results: We assess four methods for estimation of the marginal likelihoods required for computing Bayes factors. The Prior Arithmetic Mean estimator, the Posterior Harmonic Mean estimator, the Annealed Importance Sampling and the Annealing-Melting Integration methods are investigated and compared on a typical case study in Systems Biology. This allows us to understand the stability of the analysis results and make reliable judgements in uncertain context. We investigate the variance of Bayes factor estimates, and highlight the stability of the Annealed Importance Sampling and the Annealing-Melting Integration methods for the purposes of comparing nonlinear models. Availability: Models used in this study are available in SBML format as the supplementary material to this article. Contact: [email protected] Supplementary information: Supplementary data are available at Bioinformatics online.

Journal ArticleDOI
TL;DR: This article presents a series of objections to Bayesian inference, written in the voice of a hypothetical anti-Bayesian statistician to elicit elaborations and extensions of these and other arguments from non-Bayesians and responses from Bayesians who might have dieren t perspectives on these is- sues.
Abstract: Bayesian inference is one of the more controversial approaches to statistics. The fundamental objections to Bayesian methods are twofold: on one hand, Bayesian methods are presented as an automatic inference engine, and this raises suspicion in anyone with applied experience. The second objection to Bayes comes from the opposite direction and addresses the subjective strand of Bayesian inference. This article presents a series of objections to Bayesian inference, written in the voice of a hypothetical anti-Bayesian statistician. The article is intended to elicit elaborations and extensions of these and other arguments from non-Bayesians and responses from Bayesians who might have dieren t perspectives on these is- sues.

Journal ArticleDOI
TL;DR: This paper introduces a multivariate Bayesian scheme to decode or recognise brain states from neuroimages, and reduces the problem to the same form used in Gaussian process modelling, which affords a generic and efficient scheme for model optimisation and evaluating model evidence.

Journal ArticleDOI
TL;DR: It is found that proposals producing topology changes as a side effect of branch length changes (LOCAL and Continuous Change) consistently perform worse than those involving stochastic branch rearrangements (nearest neighbor interchange, subtree pruning and regrafting, tree bisection and reconnection, or subtree swapping).
Abstract: The main limiting factor in Bayesian MCMC analysis of phylogeny is typically the efficiency with which topology proposals sample tree space. Here we evaluate the performance of seven different proposal mechanisms, including most of those used in current Bayesian phylogenetics software. We sampled 12 empirical nucleotide data sets--ranging in size from 27 to 71 taxa and from 378 to 2,520 sites--under difficult conditions: short runs, no Metropolis-coupling, and an oversimplified substitution model producing difficult tree spaces (Jukes Cantor with equal site rates). Convergence was assessed by comparison to reference samples obtained from multiple Metropolis-coupled runs. We find that proposals producing topology changes as a side effect of branch length changes (LOCAL and Continuous Change) consistently perform worse than those involving stochastic branch rearrangements (nearest neighbor interchange, subtree pruning and regrafting, tree bisection and reconnection, or subtree swapping). Among the latter, moves that use an extension mechanism to mix local with more distant rearrangements show better overall performance than those involving only local or only random rearrangements. Moves with only local rearrangements tend to mix well but have long burn-in periods, whereas moves with random rearrangements often show the reverse pattern. Combinations of moves tend to perform better than single moves. The time to convergence can be shortened considerably by starting with a good tree, but this comes at the cost of compromising convergence diagnostics based on overdispersed starting points. Our results have important implications for developers of Bayesian MCMC implementations and for the large group of users of Bayesian phylogenetics software.

Journal ArticleDOI
TL;DR: The approximate Bayes estimators obtained under the assumptions of non-informative priors, are compared with the maximum likelihood estimators using Monte Carlo simulations.

Journal ArticleDOI
TL;DR: The assessment of prior probability, and how to combine the various sources of data into a statistically valid integrated assessment with a posterior probability of pathogenicity are discussed, and the use of a two‐component mixture model is proposed.
Abstract: Genetic testing often results in the finding of a variant whose clinical significance is unknown. A number of different approaches have been employed in the attempt to classify such variants. For some variants, case-control, segregation, family history, or other statistical studies can provide strong evidence of direct association with cancer risk. For most variants, other evidence is available that relates to properties of the protein or gene sequence. In this work we propose a Bayesian method for assessing the likelihood that a variant is pathogenic. We discuss the assessment of prior probability, and how to combine the various sources of data into a statistically valid integrated assessment with a posterior probability of pathogenicity. In particular, we propose the use of a two-component mixture model to integrate these various sources of data and to estimate the parameters related to sensitivity and specificity of specific kinds of evidence. Further, we discuss some of the issues involved in this process and the assumptions that underpin many of the methods used in the evaluation process.

Proceedings ArticleDOI
14 Oct 2008
TL;DR: This paper shows how Gaussian process models can be integrated into other Bayes filters, namely particle filters and extended Kalman filters, and provides a complexity analysis of these filters and evaluates the alternative techniques using data collected with an autonomous micro-blimp.
Abstract: Bayesian filtering is a general framework for recursively estimating the state of a dynamical system. The most common instantiations of Bayes filters are Kalman filters (extended and unscented) and particle filters. Key components of each Bayes filter are probabilistic prediction and observation models. Recently, Gaussian processes have been introduced as a non-parametric technique for learning such models from training data. In the context of unscented Kalman filters, these models have been shown to provide estimates that can be superior to those achieved with standard, parametric models. In this paper we show how Gaussian process models can be integrated into other Bayes filters, namely particle filters and extended Kalman filters. We provide a complexity analysis of these filters and evaluate the alternative techniques using data collected with an autonomous micro-blimp.

Journal ArticleDOI
TL;DR: It is established that under certain assumptions, the proposed Bayes' recursion reduces to the cardinalized probability hypothesis density (CPHD) recursion for a single target.
Abstract: This paper presents a novel and mathematically rigorous Bayes' recursion for tracking a target that generates multiple measurements with state dependent sensor field of view and clutter. Our Bayesian formulation is mathematically well-founded due to our use of a consistent likelihood function derived from random finite set theory. It is established that under certain assumptions, the proposed Bayes' recursion reduces to the cardinalized probability hypothesis density (CPHD) recursion for a single target. A particle implementation of the proposed recursion is given. Under linear Gaussian and constant sensor field of view assumptions, an exact closed-form solution to the proposed recursion is derived, and efficient implementations are given. Extensions of the closed-form recursion to accommodate mild nonlinearities are also given using linearization and unscented transforms.

Journal ArticleDOI
TL;DR: Bayesian methods have proven to be vastly superior to more traditional statistical tools, offering the advantage of higher efficiency and of a consistent conceptual basis for dealing with the problem of induction in the presence of uncertainty as discussed by the authors.
Abstract: The application of Bayesian methods in cosmology and astrophysics has flourished over the past decade, spurred by data sets of increasing size and complexity. In many respects, Bayesian methods have proven to be vastly superior to more traditional statistical tools, offering the advantage of higher efficiency and of a consistent conceptual basis for dealing with the problem of induction in the presence of uncertainty. This trend is likely to continue in the future, when the way we collect, manipulate and analyse observations and compare them with theoretical models will assume an even more central role in cosmology. This review is an introduction to Bayesian methods in cosmology and astrophysics and recent results in the field. I first present Bayesian probability theory and its conceptual underpinnings, Bayes' Theorem and the role of priors. I discuss the problem of parameter inference and its general solution, along with numerical techniques such as Monte Carlo Markov Chain methods. I then review the theory and application of Bayesian model comparison, discussing the notions of Bayesian evidence and effective model complexity, and how to compute and interpret those quantities. Recent developments in cosmological parameter extraction and Bayesian cosmological model building are summarized, highlighting the challenges that lie ahead.

Proceedings ArticleDOI
01 Mar 2008
TL;DR: In this article, the relevance vector machine (RVM) and particle filter (PF) are used for model identification, and the PF framework uses the learnt model, statistical estimates of noise and anticipated operational conditions to provide estimates of remaining useful life (RUL) in the form of a probability density function (PDF).
Abstract: Uncertainty management has always been the key hurdle faced by diagnostics and prognostics algorithms. A Bayesian treatment of this problem provides an elegant and theoretically sound approach to the modern Condition-Based Maintenance (CBM)/Prognostic Health Management (PHM) paradigm. The application of the Bayesian techniques to regression and classification in the form of Relevance Vector Machine (RVM), and to state estimation as in Particle Filters (PF), provides a powerful tool to integrate the diagnosis and prognosis of battery health. The RVM, which is a Bayesian treatment of the Support Vector Machine (SVM), is used for model identification, while the PF framework uses the learnt model, statistical estimates of noise and anticipated operational conditions to provide estimates of remaining useful life (RUL) in the form of a probability density function (PDF). This type of prognostics generates a significant value addition to the management of any operation involving electrical systems.

Journal ArticleDOI
TL;DR: The predictive probability design is efficient and remains robust in controlling type I and type II error rates when the trial conduct deviates from the original design.
Abstract: Background Two- or three-stage designs are commonly used in phase II cancer clinical trials. These designs possess good frequentist properties and allow early termination of the trial when the interim data indicate that the experimental regimen is inefficacious. The rigid study design, however, can be difficult to follow exactly because the response has to be evaluated at prespecified fixed number of patients.Purpose Our goal is to develop an efficient and flexible design that possesses desirable statistical properties.Methods A flexible design based on Bayesian predictive probability and the minimax criterion is constructed. A three-dimensional search algorithm is implemented to determine the design parameters.Results The new design controls type I and type II error rates, and allows continuous monitoring of the trial outcome. Consequently, under the null hypothesis when the experimental treatment is not efficacious, the design is more efficient in stopping the trial earlier, which results in a smaller e...

Journal ArticleDOI
TL;DR: In this article, the potential of alternative fully-Bayes methods, which instead margin out the hyperparameters with respect to prior distributions, is studied, and several structured prior formulations are considered for which fully Bayes selection and estimation methods are obtained.

Journal ArticleDOI
TL;DR: An algorithm is presented that provides the one-dimensional subspace, where the Bayes error is minimized for the C class problem with homoscedastic Gaussian distributions, and can be used to improve upon the outcomes provided by existing algorithms and derive a low-computational cost, linear approximation.
Abstract: We present an algorithm that provides the one-dimensional subspace, where the Bayes error is minimized for the C class problem with homoscedastic Gaussian distributions. Our main result shows that the set of possible one-dimensional spaces v, for which the order of the projected class means is identical, defines a convex region with associated convex Bayes error function g(v). This allows for the minimization of the error function using standard convex optimization algorithms. Our algorithm is then extended to the minimization of the Bayes error in the more general case of heteroscedastic distributions. This is done by means of an appropriate kernel mapping function. This result is further extended to obtain the d dimensional solution for any given d by iteratively applying our algorithm to the null space of the (d - l)-dimensional solution. We also show how this result can be used to improve upon the outcomes provided by existing algorithms and derive a low-computational cost, linear approximation. Extensive experimental validations are provided to demonstrate the use of these algorithms in classification, data analysis and visualization.

Journal ArticleDOI
TL;DR: The recently developed Bayesian wavelet‐based functional mixed model methodology is applied to analyze MALDI‐TOF mass spectrometry proteomic data to identify spectral regions that are differentially expressed across experimental conditions, in a way that takes both statistical and clinical significance into account and controls the Bayesian false discovery rate to a prespecified level.
Abstract: In this article, we apply the recently developed Bayesian wavelet-based functional mixed model methodology to analyze MALDI-TOF mass spectrometry proteomic data. By modeling mass spectra as functions, this approach avoids reliance on peak detection methods. The flexibility of this framework in modeling nonparametric fixed and random effect functions enables it to model the effects of multiple factors simultaneously, allowing one to perform inference on multiple factors of interest using the same model fit, while adjusting for clinical or experimental covariates that may affect both the intensities and locations of peaks in the spectra. For example, this provides a straightforward way to account for systematic block and batch effects that characterize these data. From the model output, we identify spectral regions that are differentially expressed across experimental conditions, in a way that takes both statistical and clinical significance into account and controls the Bayesian false discovery rate to a prespecified level. We apply this method to two cancer studies.

Journal ArticleDOI
TL;DR: This work applies the Bayesian hierarchical model to two novel fMRI data sets: one considering inhibitory control in cocaine-dependent men and the second considering verbal memory in subjects at high risk for Alzheimer's disease.

Journal ArticleDOI
TL;DR: This study experimented with four methods of treating missing values in a clinical data set and showed that in most cases the classifiers trained using the explicit missing value treatments performed better, suggesting that information may exist in "missingness" itself.

Book ChapterDOI
01 Jan 2008
TL;DR: The nature, relevance and applicability of Bayesian Network theory for issues of advanced computability form the core of the current discussion.
Abstract: Bayesian Networks are an important area of research and application within the domain of Artificial Intelligence. This paper explores the nature and implications for Bay esian Networks beginning with an overview and comparison of inferential statistics with Bayes’ Theorem. The nature, relevance and applicability of Bayesian Network theory for issues of advanced computability form the core of the current discussion. A number of current applications using Bayesian networks are examined. The paper concludes with a brief discussion of the appropriateness and limitations of Bayesian Networks for human-computer interaction and automated learning.