Showing papers on "Bayes' theorem published in 2007"

The Bayesian Choice: From Decision Theoretic Foundations to Computational Implementation

Abstract: Winner of the 2004 DeGroot Prize This paperback edition, a reprint of the 2001 edition, is a graduate-level textbook that introduces Bayesian statistics and decision theory. It covers both the basic ideas of statistical theory, and also some of the more modern and advanced topics of Bayesian statistics such as complete class theorems, the Stein effect, Bayesian model choice, hierarchical and empirical Bayes modeling, Monte Carlo integration including Gibbs sampling, and other MCMC techniques. It was awarded the 2004 DeGroot Prize by the International Society for Bayesian Analysis (ISBA) for setting "a new standard for modern textbooks dealing with Bayesian methods, especially those using MCMC techniques, and that it is a worthy successor to DeGroot's and Berger's earlier texts".

Bayesian inference of epistatic interactions in case-control studies

Abstract: Epistatic interactions among multiple genetic variants in the human genome may be important in determining individual susceptibility to common diseases. Although some existing computational methods for identifying genetic interactions have been effective for small-scale studies, we here propose a method, denoted 'bayesian epistasis association mapping' (BEAM), for genome-wide case-control studies. BEAM treats the disease-associated markers and their interactions via a bayesian partitioning model and computes, via Markov chain Monte Carlo, the posterior probability that each marker set is associated with the disease. Testing this on an age-related macular degeneration genome-wide association data set, we demonstrate that the method is significantly more powerful than existing approaches and that genome-wide case-control epistasis mapping with many thousands of markers is both computationally and statistically feasible.

A Bayesian Measure of the Probability of False Discovery in Genetic Epidemiology Studies

Abstract: In light of the vast amounts of genomic data that are now being generated, we propose a new measure, the Bayesian false-discovery probability (BFDP), for assessing the noteworthiness of an observed association. BFDP shares the ease of calculation of the recently proposed false-positive report probability (FPRP) but uses more information, has a noteworthy threshold defined naturally in terms of the costs of false discovery and nondiscovery, and has a sound methodological foundation. In addition, in a multiple-testing situation, it is straightforward to estimate the expected numbers of false discoveries and false nondiscoveries. We provide an in-depth discussion of FPRP, including a comparison with the q value, and examine the empirical behavior of these measures, along with BFDP, via simulation. Finally, we use BFDP to assess the association between 131 single-nucleotide polymorphisms and lung cancer in a case-control study.

Multiple Testing Procedures with Applications to Genomics

Abstract: Multiple Hypothesis Testing.- Test Statistics Null Distribution.- Overview of Multiple Testing Procedures.- Single-Step Multiple Testing Procedures for Controlling General Type I Error Rates, ?(Fvn).- Step-Down Multiple Testing Procedures for Controlling the Family-Wise Error Rate.- Augmentation Multiple Testing Procedures for Controlling Generalized Tail Probability Error Rates.- Resampling-Based Empirical Bayes multiple Testing Procedures for Controlling Generalized Tail Probability Error Rates.- Simulation Studies: Assessment of Test Statistics Null Distributions.- Identification of Differentially Expressed and Co-Expressed Genes in High-Throughput Gene Expression Experiments.- Multiple Tests of Association with Biological Annotation Metadata.- HIV-1 Sequence Variation and Viral Replication Capacity.- Genetic Mapping of Complex Human Traits Using Single Nucleotide Polymorphisms: The ObeLinks Project.- Software Implementation.

Probabilistic Quantitative Precipitation Forecasting Using Bayesian Model Averaging

Abstract: Bayesian model averaging (BMA) is a statistical way of postprocessing forecast ensembles to create predictive probability density functions (PDFs) for weather quantities. It represents the predictive PDF as a weighted average of PDFs centered on the individual bias-corrected forecasts, where the weights are posterior probabilities of the models generating the forecasts and reflect the forecasts’ relative contributions to predictive skill over a training period. It was developed initially for quantities whose PDFs can be approximated by normal distributions, such as temperature and sea level pressure. BMA does not apply in its original form to precipitation, because the predictive PDF of precipitation is nonnormal in two major ways: it has a positive probability of being equal to zero, and it is skewed. In this study BMA is extended to probabilistic quantitative precipitation forecasting. The predictive PDF corresponding to one ensemble member is a mixture of a discrete component at zero and a gam...

A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data.

Abstract: In the analysis of data generated by change-point processes, one critical challenge is to determine the number of change-points. The classic Bayes information criterion (BIC) statistic does not work well here because of irregularities in the likelihood function. By asymptotic approximation of the Bayes factor, we derive a modified BIC for the model of Brownian motion with changing drift. The modified BIC is similar to the classic BIC in the sense that the first term consists of the log likelihood, but it differs in the terms that penalize for model dimension. As an example of application, this new statistic is used to analyze array-based comparative genomic hybridization (array-CGH) data. Array-CGH measures the number of chromosome copies at each genome location of a cell sample, and is useful for finding the regions of genome deletion and amplification in tumor cells. The modified BIC performs well compared to existing methods in accurately choosing the number of regions of changed copy number. Unlike existing methods, it does not rely on tuning parameters or intensive computing. Thus it is impartial and easier to understand and to use.

Understanding diagnostic tests 2: likelihood ratios, pre- and post-test probabilities and their use in clinical practice

Abstract: The sensitivity and specificity of a test cannot be used to estimate probability of disease in individual patients. They can, however, be combined into a single measure called the likelihood ratio which is, clinically, more useful than sensitivity or specificity. Likelihood ratios provide a summary of how many times more (or less) likely patients with a disease are to have a particular result than patients without the disease. Using the principles of the Bayes theorem, likelihood ratios can be used in conjunction with pre-test probability of disease to estimate an individual’s post-test probability of disease, that is his or her chance of having disease once the result of a test is known. The Fagan’s nomogram is a graphical tool which, in routine clinical practice, allows one to combine the likelihood ratio of a test with a patient’s pre-test probability of disease to estimate post-test probability. Conclusion: Likelihood ratios summarize information about a diagnostic test by combining sensitivity and specificity. The Fagan’s nomogram is a useful and convenient graphical tool that allows likelihood ratios to be used in conjunction with a patient’s pre-test probability of disease to estimate the post-test probability of disease.

Size, power and false discovery rates

Abstract: Modern scientific technology has provided a new class of large-scale simultaneous inference problems, with thousands of hypothesis tests to consider at the same time. Microarrays epitomize this type of technology, but similar situations arise in proteomics, spectroscopy, imaging, and social science surveys. This paper uses false discovery rate methods to carry out both size and power calculations on large-scale problems. A simple empirical Bayes approach allows the false discovery rate (fdr) analysis to proceed with a minimum of frequentist or Bayesian modeling assumptions. Closed-form accuracy formulas are derived for estimated false discovery rates, and used to compare different methodologies: local or tail-area fdr's, theoretical, permutation, or empirical null hypothesis estimates. Two microarray data sets as well as simulations are used to evaluate the methodology, the power diagnostics showing why nonnull cases might easily fail to appear on a list of ``significant'' discoveries.

Size, power and false discovery rates

Abstract: Modern scientific technology has provided a new class of large-scale simultaneous inference problems, with thousands of hypothesis tests to consider at the same time. Microarrays epitomize this type of technology, but similar situations arise in proteomics, spectroscopy, imaging, and social science surveys. This paper uses false discovery rate methods to carry out both size and power calculations on large-scale problems. A simple empirical Bayes approach allows the false discovery rate (fdr) analysis to proceed with a minimum of frequentist or Bayesian modeling assumptions. Closed-form accuracy formulas are derived for estimated false discovery rates, and used to compare different methodologies: local or tail-area fdr’s, theoretical, permutation, or empirical null hypothesis estimates. Two microarray data sets as well as simulations are used to evaluate the methodology, the power diagnostics showing why nonnull cases might easily fail to appear on a list of “significant” discoveries.

Inference of Population Structure Under a Dirichlet Process Model

Abstract: Inferring population structure from genetic data sampled from some number of individuals is a formidable statistical problem. One widely used approach considers the number of populations to be fixed and calculates the posterior probability of assigning individuals to each population. More recently, the assignment of individuals to populations and the number of populations have both been considered random variables that follow a Dirichlet process prior. We examined the statistical behavior of assignment of individuals to populations under a Dirichlet process prior. First, we examined a best-case scenario, in which all of the assumptions of the Dirichlet process prior were satisfied, by generating data under a Dirichlet process prior. Second, we examined the performance of the method when the genetic data were generated under a population genetics model with symmetric migration between populations. We examined the accuracy of population assignment using a distance on partitions. The method can be quite accurate with a moderate number of loci. As expected, inferences on the number of populations are more accurate when θ = 4Neu is large and when the migration rate (4Nem) is low. We also examined the sensitivity of inferences of population structure to choice of the parameter of the Dirichlet process model. Although inferences could be sensitive to the choice of the prior on the number of populations, this sensitivity occurred when the number of loci sampled was small; inferences are more robust to the prior on the number of populations when the number of sampled loci is large. Finally, we discuss several methods for summarizing the results of a Bayesian Markov chain Monte Carlo (MCMC) analysis of population structure. We develop the notion of the mean population partition, which is the partition of individuals to populations that minimizes the squared partition distance to the partitions sampled by the MCMC algorithm.

A Primer on Learning in Bayesian Networks for Computational Biology

Abstract: Bayesian networks (BNs) provide a neat and compact representation for expressing joint probability distributions (JPDs) and for inference. They are becoming increasingly important in the biological sciences for the tasks of inferring cellular networks [1], modelling protein signalling pathways [2], systems biology, data integration [3], classification [4], and genetic data analysis [5]. The representation and use of probability theory makes BNs suitable for combining domain knowledge and data, expressing causal relationships, avoiding overfitting a model to training data, and learning from incomplete datasets. The probabilistic formalism provides a natural treatment for the stochastic nature of biological systems and measurements. This primer aims to introduce BNs to the computational biologist, focusing on the concepts behind methods for learning the parameters and structure of models, at a time when they are becoming the machine learning method of choice. There are many applications in biology where we wish to classify data; for example, gene function prediction. To solve such problems, a set of rules are required that can be used for prediction, but often such knowledge is unavailable, or in practice there turn out to be many exceptions to the rules or so many rules that this approach produces poor results. Machine learning approaches often produce better results, where a large number of examples (the training set) is used to adapt the parameters of a model that can then be used for performing predictions or classifications on data. There are many different types of models that may be required and many different approaches to training the models, each with its pros and cons. An excellent overview of the topic can be found in [6] and [7]. Neural networks, for example, are often able to learn a model from training data, but it is often difficult to extract information about the model, which with other methods can provide valuable insights into the data or problem being solved. A common problem in machine learning is overfitting, where the learned model is too complex and generalises poorly to unseen data. Increasing the size of the training dataset may reduce this; however, this assumes more training data is readily available, which is often not the case. In addition, often it is important to determine the uncertainty in the learned model parameters or even in the choice of model. This primer focuses on the use of BNs, which offer a solution to these issues. The use of Bayesian probability theory provides mechanisms for describing uncertainty and for adapting the number of parameters to the size of the data. Using a graphical representation provides a simple way to visualise the structure of a model. Inspection of models can provide valuable insights into the properties of the data and allow new models to be produced.

Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models

Abstract: SUMMARY The problem of evaluating the goodness of the predictive distributions of hierarchical Bayesian and empirical Bayes models is investigated. A Bayesian predictive information criterion is proposed as an estimator of the posterior mean of the expected loglikelihood of the predictive distribution when the specified family of probability distributions does not contain the true distribution. The proposed criterion is developed by correcting the asymptotic bias of the posterior mean of the loglikelihood as an estimator of its expected loglikelihood. In the evaluation of hierarchical Bayesian models with random effects, regardless of our parametric focus, the proposed criterion considers the bias correction of the posterior mean of the marginal loglikelihood because it requires a consistent parameter estimator. The use of the bootstrap in model evaluation is also discussed.

Accurate ranking of differentially expressed genes by a distribution-free shrinkage approach

Abstract: High-dimensional case-control analysis is encountered in many different settings in genomics. In order to rank genes accordingly, many different scores have been proposed, ranging from ad hoc modifications of the ordinary t statistic to complicated hierarchical Bayesian models. Here, we introduce the "shrinkage t" statistic that is based on a novel and model-free shrinkage estimate of the variance vector across genes. This is derived in a quasi-empirical Bayes setting. The new rank score is fully automatic and requires no specification of parameters or distributions. It is computationally inexpensive and can be written analytically in closed form. Using a series of synthetic and three real expression data we studied the quality of gene rankings produced by the "shrinkage t" statistic. The new score consistently leads to highly accurate rankings for the complete range of investigated data sets and all considered scenarios for across-gene variance structures.

Bayesian sensitivity analysis for unmeasured confounding in observational studies

Abstract: We consider Bayesian sensitivity analysis for unmeasured confounding in observational studies where the association between a binary exposure, binary response, measured confounders and a single binary unmeasured confounder can be formulated using logistic regression models. A model for unmeasured confounding is presented along with a family of prior distributions that model beliefs about a possible unknown unmeasured confounder. Simulation from the posterior distribution is accomplished using Markov chain Monte Carlo. Because the model for unmeasured confounding is not identifiable, standard large-sample theory for Bayesian analysis is not applicable. Consequently, the impact of different choices of prior distributions on the coverage probability of credible intervals is unknown. Using simulations, we investigate the coverage probability when averaged with respect to various distributions over the parameter space. The results indicate that credible intervals will have approximately nominal coverage probability, on average, when the prior distribution used for sensitivity analysis approximates the sampling distribution of model parameters in a hypothetical sequence of observational studies. We motivate the method in a study of the effectiveness of beta blocker therapy for treatment of heart failure.

An empirical Bayes method for estimating epistatic effects of quantitative trait loci.

Abstract: The genetic variance of a quantitative trait is often controlled by the segregation of multiple interacting loci. Linear model regression analysis is usually applied to estimating and testing effects of these quantitative trait loci (QTL). Including all the main effects and the effects of interaction (epistatic effects), the dimension of the linear model can be extremely high. Variable selection via stepwise regression or stochastic search variable selection (SSVS) is the common procedure for epistatic effect QTL analysis. These methods are computationally intensive, yet they may not be optimal. The LASSO (least absolute shrinkage and selection operator) method is computationally more efficient than the above methods. As a result, it has been widely used in regression analysis for large models. However, LASSO has never been applied to genetic mapping for epistatic QTL, where the number of model effects is typically many times larger than the sample size. In this study, we developed an empirical Bayes method (E-BAYES) to map epistatic QTL under the mixed model framework. We also tested the feasibility of using LASSO to estimate epistatic effects, examined the fully Bayesian SSVS, and reevaluated the penalized likelihood (PENAL) methods in mapping epistatic QTL. Simulation studies showed that all the above methods performed satisfactorily well. However, E-BAYES appears to outperform all other methods in terms of minimizing the mean-squared error (MSE) with relatively short computing time. Application of the new method to real data was demonstrated using a barley dataset.

Bayesian perspectives for epidemiological research. II. Regression analysis

Abstract: This article describes extensions of the basic Bayesian methods using data priors to regression modelling, including hierarchical (multilevel) models. These methods provide an alternative to the parsimony-oriented approach of frequentist regression analysis. In particular, they replace arbitrary variableselection criteria by prior distributions, and by doing so facilitate realistic use of imprecise but important prior information. They also allow Bayesian analyses to be conducted using standard regression packages; one need only be able to add variables and records to the data set. The methods thus facilitate the use of Bayesian solutions to problems of sparse data, multiple comparisons, subgroup analyses and study bias. Because these solutions have a frequentist interpretation as ‘shrinkage’ (penalized) estimators, the methods can also be viewed as a means of implementing shrinkage approaches to multiparameter problems.

Bayes Linear Statistics: Theory and Methods

Abstract: The text provides a thorough coverage of Bayes linear analysis, from the development of the basic language to the collection of algebraic results needed for efficient implementation, with detailed practical examples. The book covers: The importance of partial prior specifications for complex problems where it is difficult to supply a meaningful full prior probability specification. Simple ways to use partial prior specifications to adjust beliefs, given observations. Interpretative and diagnostic tools to display the implications of collections of belief statements, and to make stringent comparisons between expected and actual observations. General approaches to statistical modelling based upon partial exchangeability judgements. Bayes linear graphical models to represent and display partial belief specifications, organize computations, and display the results of analyses.

MIDAS: Stata module for meta-analytical integration of diagnostic test accuracy studies

Abstract: midas is a user-written command for idiot-proof implementation of some of the contemporary statistical methods for meta-analysis of binary diagnostic test accuracy. Primary data synthesis is performed within the bivariate mixed-effects logistic regression modeling framework. Likelihood-based estimation is by adaptive gaussian quadrature using xtmelogit (Stata release 10) with post-estimation procedures for model diagnostics and empirical Bayes predictions. Average sensitivity and specificity (optionally depicted in SROC space with or without confidence and prediction regions), and their derivative likelihood and odds ratios are calculated from the maximum likelihood estimates. midas facilitates exploratory analysis of heterogeneity, threshold-related variability, methodological quality bias, publication and other precision-related biases. Bayes' nomograms, likelihood-ratio matrices, and probability modifying plots may be derived and used to guide patient-based diagnostic decision making. A dataset of studies evaluating axillary staging performance of positron emission tomography in breast cancer patients is provided for illustration of the omnibus capabilities of midas.

A Collapsed Variational Bayesian Inference Algorithm for Latent Dirichlet Allocation

Abstract: Latent Dirichlet allocation (LDA) is a Bayesian network that has recently gained much popularity in applications ranging from document modeling to computer vision Due to the large scale nature of these applications, current inference procedures like variational Bayes and Gibbs sampling have been found lacking In this paper we propose the collapsed variational Bayesian inference algorithm for LDA, and show that it is computationally efficient, easy to implement and significantly more accurate than standard variational Bayesian inference for LDA

Bayesian Time-Series Model for Short-Term Traffic Flow Forecasting

Abstract: The seasonal autoregressive integrated moving average (SARIMA) model is one of the popular univariate time-series models in the field of short-term traffic flow forecasting. The parameters of the SARIMA model are commonly estimated using classical (maximum likelihood estimate and/or least-squares estimate) methods. In this paper, instead of using classical inference the Bayesian method is employed to estimate the parameters of the SARIMA model considered for modeling. In Bayesian analysis the Markov chain Monte Carlo method is used to solve the posterior integration problem in high dimension. Each of the estimated parameters from the Bayesian method has a probability density function conditional to the observed traffic volumes. The forecasts from the Bayesian model can better match the traffic behavior of extreme peaks and rapid fluctuation. Similar to the estimated parameters, each forecast has a probability density curve with the maximum probable value as the point forecast. Individual probability density curves provide a time-varying prediction interval unlike the constant prediction interval from the classical inference. The time-series data used for fitting the SARIMA model are obtained from a certain junction in the city center of Dublin.

Feature-Preserving MRI Denoising: A Nonparametric Empirical Bayes Approach

Abstract: This paper presents a novel method for Bayesian denoising of magnetic resonance (MR) images that bootstraps itself by inferring the prior, i.e., the uncorrupted-image statistics, from the corrupted input data and the knowledge of the Rician noise model. The proposed method relies on principles from empirical Bayes (EB) estimation. It models the prior in a nonparametric Markov random field (MRF) framework and estimates this prior by optimizing an information-theoretic metric using the expectation-maximization algorithm. The generality and power of nonparametric modeling, coupled with the EB approach for prior estimation, avoids imposing ill-fitting prior models for denoising. The results demonstrate that, unlike typical denoising methods, the proposed method preserves most of the important features in brain MR images. Furthermore, this paper presents a novel Bayesian-inference algorithm on MRFs, namely iterated conditional entropy reduction (ICER). This paper also extends the application of the proposed method for denoising diffusion-weighted MR images. Validation results and quantitative comparisons with the state of the art in MR-image denoising clearly depict the advantages of the proposed method.

Assessing Uncertainty in Urban Simulations Using Bayesian Melding

Abstract: We develop a method for assessing uncertainty about quantities of interest using urban simulation models The method is called Bayesian melding, and extends a previous method developed for macrolevel deterministic simulation models to agent-based stochastic models It encodes all the available information about model inputs and outputs in terms of prior probability distributions and likelihoods, and uses Bayes’s theorem to obtain the resulting posterior distribution of any quantity of interest that is a function of model inputs and/or outputs It is Monte Carlo based, and quite easy to implement We applied it to the projection of future household numbers by traffic activity zone in Eugene-Springfield, Oregon, using the UrbanSim model developed at the University of Washington We compared it with a simpler method that uses repeated runs of the model with fixed estimated inputs We found that the simple repeated runs method gave distributions of quantities of interest that were too narrow, while Bayesian melding gave well calibrated uncertainty statements � 2006 Elsevier Ltd All rights reserved

On universal prediction and Bayesian confirmation

Abstract: The Bayesian framework is a well-studied and successful framework for inductive reasoning, which includes hypothesis testing and confirmation, parameter estimation, sequence prediction, classification, and regression. But standard statistical guidelines for choosing the model class and prior are not always available or can fail, in particular in complex situations. Solomonoff completed the Bayesian framework by providing a rigorous, unique, formal, and universal choice for the model class and the prior. I discuss in breadth how and in which sense universal (non-i.i.d.) sequence prediction solves various (philosophical) problems of traditional Bayesian sequence prediction. I show that Solomonoff’s model possesses many desirable properties: strong total and future bounds, and weak instantaneous bounds, and, in contrast to most classical continuous prior densities, it has no zero p(oste)rior problem, i.e. it can confirm universal hypotheses, is reparametrization and regrouping invariant, and avoids the old-evidence and updating problem. It even performs well (actually better) in non-computable environments.

Tree growth inference and prediction from diameter censuses and ring widths

Abstract: Estimation of tree growth is based on sparse observations of tree diameter, ring widths, or increments read from a dendrometer. From annual measurements on a few trees (e.g., increment cores) or sporadic measurements from many trees (e.g., diameter censuses on mapped plots), relationships with resources, tree size, and climate are extrapolated to whole stands. There has been no way to formally integrate different types of data and problems of estimation that result from (1) multiple sources of observation error, which frequently result in impossible estimates of negative growth, (2) the fact that data are typically sparse (a few trees or a few years), whereas inference is needed broadly (many trees over many years), (3) the fact that some unknown fraction of the variance is shared across the population, and (4) the fact that growth rates of trees within competing stands are not independent. We develop a hierarchical Bayes state space model for tree growth that addresses all of these challenges, allowing for formal inference that is consistent with the available data and the assumption that growth is nonnegative. Prediction follows directly, incorporating the full uncertainty from inference with scenarios for ''filling the gaps'' for past growth rates and for future conditions affecting growth. An example involving multiple species and multiple stands with tree-ring data and up to 14 years of tree census data illustrates how different levels of information at the tree and stand level contribute to inference and prediction.

Naïve Bayes Text Classifier

Abstract: Text classification algorithms, such SVM, and Naive Bayes, have been developed to build up search engines and construct spam email filters. As a simple yet powerful sample of Bayesian theorem, naive Bayes shows advantages in text classification yielding satisfactory results. In this paper, a spam email detector is developed using naive Bayes algorithm. We use pre-classified emails (priory knowledge) to train the spam email detector. With the model generated from the training step, the detector is able to decide whether an email is a spam email or an ordinary email.

Hierarchical Bayes prioritization of marker associations from a genome‐wide association scan for further investigation

Abstract: We describe a hierarchical regression modeling approach to selection of a subset of markers from the first stage of a genomewide association scan to carry forward to subsequent stages for testing on an independent set of subjects. Rather than simply selecting a subset of most significant marker-disease associations at some cutoff chosen to maximize the cost efficiency of a multistage design, we propose a prior model for the true noncentrality parameters of these associations composed of a large mass at zero and a continuous distribution of nonzero values. The prior probability of nonzero values and their prior means can be functions of various covariates characterizing each marker, such as their location relative to genes or evolutionary conserved regions, or prior linkage or association data. We propose to take the top ranked posterior expectations of the noncentrality parameters for confirmation in later stages of a genomewide scan. The statistical performance of this approach is compared with the traditional p-value ranking by simulation studies. We show that the ranking by posterior expectations performs better at selecting the true positive association than a simple ranking of p-values if at least some of the prior covariates have predictive value.

An Efficient Bayesian Model Selection Approach for Interacting Quantitative Trait Loci Models With Many Effects

Abstract: We extend our Bayesian model selection framework for mapping epistatic QTL in experimental crosses to include environmental effects and gene–environment interactions. We propose a new, fast Markov chain Monte Carlo algorithm to explore the posterior distribution of unknowns. In addition, we take advantage of any prior knowledge about genetic architecture to increase posterior probability on more probable models. These enhancements have significant computational advantages in models with many effects. We illustrate the proposed method by detecting new epistatic and gene–sex interactions for obesity-related traits in two real data sets of mice. Our method has been implemented in the freely available package R/qtlbim (http://www.qtlbim.org) to facilitate the general usage of the Bayesian methodology for genomewide interacting QTL analysis.

The Bayesian ARTMAP

Abstract: In this paper, we modify the fuzzy ARTMAP (FA) neural network (NN) using the Bayesian framework in order to improve its classification accuracy while simultaneously reduce its category proliferation. The proposed algorithm, called Bayesian ARTMAP (BA), preserves the FA advantages and also enhances its performance by the following: (1) representing a category using a multidimensional Gaussian distribution, (2) allowing a category to grow or shrink, (3) limiting a category hypervolume, (4) using Bayes' decision theory for learning and inference, and (5) employing the probabilistic association between every category and a class in order to predict the class. In addition, the BA estimates the class posterior probability and thereby enables the introduction of loss and classification according to the minimum expected loss. Based on these characteristics and using synthetic and 20 real-world databases, we show that the BA outperformes the FA, either trained for one epoch or until completion, with respect to classification accuracy, sensitivity to statistical overlapping, learning curves, expected loss, and category proliferation.