Showing papers on "Bayes' theorem published in 2003"

Multitarget Bayes filtering via first-order multitarget moments

Abstract: The theoretically optimal approach to multisensor-multitarget detection, tracking, and identification is a suitable generalization of the recursive Bayes nonlinear filter. Even in single-target problems, this optimal filter is so computationally challenging that it must usually be approximated. Consequently, multitarget Bayes filtering will never be of practical interest without the development of drastic but principled approximation strategies. In single-target problems, the computationally fastest approximate filtering approach is the constant-gain Kalman filter. This filter propagates a first-order statistical moment - the posterior expectation - in the place of the posterior distribution. The purpose of this paper is to propose an analogous strategy for multitarget systems: propagation of a first-order statistical moment of the multitarget posterior. This moment, the probability hypothesis density (PHD), is the function whose integral in any region of state space is the expected number of targets in that region. We derive recursive Bayes filter equations for the PHD that account for multiple sensors, nonconstant probability of detection, Poisson false alarms, and appearance, spawning, and disappearance of targets. We also show that the PHD is a best-fit approximation of the multitarget posterior in an information-theoretic sense.

Reasoning about Uncertainty

Abstract: In order to deal with uncertainty intelligently, we need to be able to represent it and reason about it. In this book, Joseph Halpern examines formal ways of representing uncertainty and considers various logics for reasoning about it. While the ideas presented are formalized in terms of definitions and theorems, the emphasis is on the philosophy of representing and reasoning about uncertainty. Halpern surveys possible formal systems for representing uncertainty, including probability measures, possibility measures, and plausibility measures; considers the updating of beliefs based on changing information and the relation to Bayes' theorem; and discusses qualitative, quantitative, and plausibilistic Bayesian networks. This second edition has been updated to reflect Halpern's recent research. New material includes a consideration of weighted probability measures and how they can be used in decision making; analyses of the Doomsday argument and the Sleeping Beauty problem; modeling games with imperfect recall using the runs-and-systems approach; a discussion of complexity-theoretic considerations; the application of first-order conditional logic to security. Reasoning about Uncertainty is accessible and relevant to researchers and students in many fields, including computer science, artificial intelligence, economics (particularly game theory), mathematics, philosophy, and statistics.

Bayes or Bootstrap? A Simulation Study Comparing the Performance of Bayesian Markov Chain Monte Carlo Sampling and Bootstrapping in Assessing Phylogenetic Confidence

Abstract: Bayesian Markov chain Monte Carlo sampling has become increasingly popular in phylogenetics as a method for both estimating the maximum likelihood topology and for assessing nodal confidence. Despite the growing use of posterior probabilities, the relationship between the Bayesian measure of confidence and the most commonly used confidence measure in phylogenetics, the nonparametric bootstrap proportion, is poorly understood. We used computer simulation to investigate the behavior of three phylogenetic confidence methods: Bayesian posterior probabilities calculated via Markov chain Monte Carlo sampling (BMCMC-PP), maximum likelihood bootstrap proportion (ML-BP), and maximum parsimony bootstrap proportion (MP-BP). We simulated the evolution of DNA sequence on 17-taxon topologies under 18 evolutionary scenarios and examined the performance of these methods in assigning confidence to correct monophyletic and incorrect monophyletic groups, and we examined the effects of increasing character number on support value. BMCMC-PP and ML-BP were often strongly correlated with one another but could provide substantially different estimates of support on short internodes. In contrast, BMCMC-PP correlated poorly with MP-BP across most of the simulation conditions that we examined. For a given threshold value, more correct monophyletic groups were supported by BMCMC-PP than by either ML-BP or MP-BP. When threshold values were chosen that fixed the rate of accepting incorrect monophyletic relationship as true at 5%, all three methods recovered most of the correct relationships on the simulated topologies, although BMCMC-PP and ML-BP performed better than MP-BP. BMCMC-PP was usually a less biased predictor of phylogenetic accuracy than either bootstrapping method. BMCMC-PP provided high support values for correct topological bipartitions with fewer characters than was needed for nonparametric bootstrap.

Bayesian analysis of genetic differentiation between populations.

Abstract: We introduce a Bayesian method for estimating hidden population substructure using multilocus molecular markers and geographical information provided by the sampling design. The joint posterior distribution of the substructure and allele frequencies of the respective populations is available in an analytical form when the number of populations is small, whereas an approximation based on a Markov chain Monte Carlo simulation approach can be obtained for a moderate or large number of populations. Using the joint posterior distribution, posteriors can also be derived for any evolutionary population parameters, such as the traditional fixation indices. A major advantage compared to most earlier methods is that the number of populations is treated here as an unknown parameter. What is traditionally considered as two genetically distinct populations, either recently founded or connected by considerable gene flow, is here considered as one panmictic population with a certain probability based on marker data and prior information. Analyses of previously published data on the Moroccan argan tree (Argania spinosa) and of simulated data sets suggest that our method is capable of estimating a population substructure, while not artificially enforcing a substructure when it does not exist. The software (BAPS) used for the computations is freely available from http://www.rni.helsinki.fi/~mjs.

A Bayesian missing value estimation method for gene expression profile data

Abstract: Motivation: Gene expression profile analyses have been used in numerous studies covering a broad range of areas in biology. When unreliable measurements are excluded, missing values are introduced in gene expression profiles. Although existing multivariate analysis methods have difficulty with the treatment of missing values, this problem has received little attention. There are many options for dealing with missing values, each of which reaches drastically different results. Ignoring missing values is the simplest method and is frequently applied. This approach, however, has its flaws. In this article, we propose an estimation method for missing values, which is based on Bayesian principal component analysis (BPCA). Although the methodology that a probabilistic model and latent variables are estimated simultaneously within the framework of Bayes inference is not new in principle, actual BPCA implementation that makes it possible to estimate arbitrary missing variables is new in terms of statistical methodology. Results: When applied to DNA microarray data from various experimental conditions, the BPCA method exhibited markedly better estimation ability than other recently proposed methods, such as singular value decomposition and K -nearest neighbors. While the estimation performance of existing methods depends on model parameters whose determination is difficult, our BPCA method is free from this difficulty. Accordingly, the BPCA method provides accurate and convenient estimation for missing values. Availability: The software is available at http://hawaii.aist

Reliability of Bayesian posterior probabilities and bootstrap frequencies in phylogenetics.

Abstract: Many empirical studies have revealed considerable differences between nonparametric bootstrapping and Bayesian posterior probabilities in terms of the support values for branches, despite claimed predictions about their approximate equivalence. We investigated this problem by simulating data, which were then analyzed by maximum likelihood bootstrapping and Bayesian phylogenetic analysis using identical models and reoptimization of parameter values. We show that Bayesian posterior probabilities are significantly higher than corresponding nonparametric bootstrap frequencies for true clades, but also that erroneous conclusions will be made more often. These errors are strongly accentuated when the models used for analyses are underparameterized. When data are analyzed under the correct model, nonparametric bootstrapping is conservative. Bayesian posterior probabilities are also conservative in this respect, but less so.

Comparison of Bayesian and Maximum Likelihood Bootstrap Measures of Phylogenetic Reliability

Abstract: 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.

Gene networks inference using dynamic Bayesian networks

Abstract: This article deals with the identification of gene regulatory networks from experimental data using a statistical machine learning approach. A stochastic model of gene interactions capable of handling missing variables is proposed. It can be described as a dynamic Bayesian network particularly well suited to tackle the stochastic nature of gene regulation and gene expression measurement. Parameters of the model are learned through a penalized likelihood maximization implemented through an extended version of EM algorithm. Our approach is tested against experimental data relative to the S.O.S. DNA Repair network of the Escherichia coli bacterium. It appears to be able to extract the main regulations between the genes involved in this network. An added missing variable is found to model the main protein of the network. Good prediction abilities on unlearned data are observed. These first results are very promising: they show the power of the learning algorithm and the ability of the model to capture gene interactions.

Inference and computation with population codes

Abstract: In the vertebrate nervous system, sensory stimuli are typically encoded through the concerted activity of large populations of neurons. Classically, these patterns of activity have been treated as encoding the value of the stimulus (e.g., the orientation of a contour), and computation has been formalized in terms of function approximation. More recently, there have been several suggestions that neural computation is akin to a Bayesian inference process, with population activity patterns representing uncertainty about stimuli in the form of probability distributions (e.g., the probability density function over the orientation of a contour). This paper reviews both approaches, with a particular emphasis on the latter, which we see as a very promising framework for future modeling and experimental work.

Modeling traffic crash-flow relationships for intersections: dispersion parameter, functional form, and bayes versus empirical bayes methods

Abstract: Statistical relationships between traffic crashes and traffic flows at roadway intersections have been extensively modeled and evaluated in recent years. The underlying assumptions adopted in the popular models for intersections are challenged. First, the assumption that the dispersion parameter is a fixed parameter across sites and time periods is challenged. Second, the mathematical limitations of some functional forms used in these models, particularly their properties at the boundaries, are examined. It is also demonstrated that, for a given data set, a large number of plausible functional forms with almost the same overall statistical goodness of fit (GOF) is possible, and an alternative class of logical formulations that may enable a richer interpretation of the data is introduced. A comparison of site estimates from the empirical Bayes and full Bayes methods is also presented. All discussions and comparisons are illustrated with a set of data collected for an urban four-legged signalized intersection in Toronto, Ontario, Canada, from 1990 to 1995. In discussing functional forms, the need for some goodness-of-logic measures, in addition to the GOF measure, is emphasized and demonstrated. Finally, analysts are advised to be mindful of the underlying assumptions adopted in the popular models, especially the assumption that the dispersion parameter is a fixed parameter, and the limitations of the functional forms used. Promising directions in which this study may be extended are also discussed.

On parametric empirical Bayes methods for comparing multiple groups using replicated gene expression profiles

Abstract: DNA microarrays provide for unprecedented large-scale views of gene expression and, as a result, have emerged as a fundamental measurement tool in the study of diverse biological systems. Statistical questions abound, but many traditional data analytic approaches do not apply, in large part because thousands of individual genes are measured with relatively little replication. Empirical Bayes methods provide a natural approach to microarray data analysis because they can significantly reduce the dimensionality of an inference problem while compensating for relatively few replicates by using information across the array. We propose a general empirical Bayes modelling approach which allows for replicate expression profiles in multiple conditions. The hierarchical mixture model accounts for differences among genes in their average expression levels, differential expression for a given gene among cell types, and measurement fluctuations. Two distinct parameterizations are considered: a model based on Gamma distributed measurements and one based on log-normally distributed measurements. False discovery rate and related operating characteristics of the methodology are assessed in a simulation study. We also show how the posterior odds of differential expression in one version of the model is related to the ratio of the arithmetic mean to the geometric mean of the two sample means. The methodology is used in a study of mammary cancer in the rat, where four distinct patterns of expression are possible. Copyright © 2003 John Wiley & Sons, Ltd.

Gene selection: a Bayesian variable selection approach

Abstract: Selection of significant genes via expression patterns is an important problem in microarray experiments. Owing to small sample size and the large number of variables (genes), the selection process can be unstable. This paper proposes a hierarchical Bayesian model for gene (variable) selection. We employ latent variables to specialize the model to a regression setting and uses a Bayesian mixture prior to perform the variable selection. We control the size of the model by assigning a prior distribution over the dimension (number of significant genes) of the model. The posterior distributions of the parameters are not in explicit form and we need to use a combination of truncated sampling and Markov Chain Monte Carlo (MCMC) based computation techniques to simulate the parameters from the posteriors. The Bayesian model is flexible enough to identify significant genes as well as to perform future predictions. The method is applied to cancer classification via cDNA microarrays where the genes BRCA1 and BRCA2 are associated with a hereditary disposition to breast cancer, and the method is used to identify a set of significant genes. The method is also applied successfully to the leukemia data.

Comparison of likelihood and bayesian methods for estimating divergence times using multiple gene loci and calibration points, with application to a radiation of cute-looking mouse lemur species

Abstract: Divergence time and substitution rate are seriously confounded in phylogenetic analysis, making it difficult to estimate divergence times when the molecular clock (rate constancy among lineages) is violated. This problem can be alleviated to some extent by analyzing multiple gene loci simultaneously and by using multiple calibration points. While different genes may have different patterns of evolutionary rate change, they share the same divergence times. Indeed, the fact that each gene may violate the molecular clock differently leads to the advantage of simultaneous analysis of multiple loci. Multiple calibration points provide the means for characterizing the local evolutionary rates on the phylogeny. In this paper, we extend previous likelihood models of local molecular clock for estimating species divergence times to accommodate multiple calibration points and multiple genes. Heterogeneity among different genes in evolutionary rate and in substitution process is accounted for by the models. We apply the likelihood models to analyze two mitochondrial protein-coding genes, cytochrome oxidase II and cytochrome b, to estimate divergence times of Malagasy mouse lemurs and related outgroups. The likelihood method is compared with the Bayes method of Thorne et al. (1998, Mol. Biol. Evol. 15:1647-1657), which uses a probabilistic model to describe the change in evolutionary rate over time and uses the Markov chain Monte Carlo procedure to derive the posterior distribution of rates and times. Our likelihood implementation has the drawbacks of failing to accommodate uncertainties in fossil calibrations and of requiring the researcher to classify branches on the tree into different rate groups. Both problems are avoided in the Bayes method. Despite the differences in the two methods, however, data partitions and model assumptions had the greatest impact on date estimation. The three codon positions have very different substitution rates and evolutionary dynamics, and assumptions in the substitution model affect date estimation in both likelihood and Bayes analyses. The results demonstrate that the separate analysis is unreliable, with dates variable among codon positions and between methods, and that the combined analysis is much more reliable. When the three codon positions were analyzed simultaneously under the most realistic models using all available calibration information, the two methods produced similar results. The divergence of the mouse lemurs is dated to be around 7-10 million years ago, indicating a surprisingly early species radiation for such a morphologically uniform group of primates.

Learning Bayesian Network Model Structure from Data

Abstract: : In this thesis I address the important problem of the determination of the structure of directed statistical models, with the widely used class of Bayesian network models as a concrete vehicle of my ideas. The structure of a Bayesian network represents a set of conditional independence relations that hold in the domain. Learning the structure of the Bayesian network model that represents a domain can reveal insights into its underlying causal structure. Moreover, it can also be used for prediction of quantities that are dif cult, expensive, or unethical to measure such as the probability of lung cancer for example based on other quantities that are easier to obtain. The contributions of this thesis include (a) an algorithm for determining the structure of a Bayesian network model from statistical independence statements; (b) a statistical independence test for continuous variables; and nally (c) a practical application of structure learning to a decision support problem, where a model learned from the database most importantly its structure is used in lieu of the database to yield fast approximate answers to count queries, surpassing in certain aspects other state-of-the-art approaches to the same problem.

On the Bayes-risk consistency of regularized boosting methods

Abstract: The probability of error of classification methods based on convex combinations of simple base classifiers by "boosting" algorithms is investigated. The main result of the paper is that certain regularized boosting algorithms provide Bayes-risk consistent classifiers under the sole assumption that the Bayes classifier may be approximated by a convex combination of the base classifiers. Nonasymptotic distribution-free bounds are also developed which offer interesting new insight into how boosting works and help explain its success in practical classification problems.

Bayesian reasoning in data analysis: a critical introduction

Abstract: Critical review and outline of the Bayesian alternative: uncertainty in physics and the usual methods of handling it a probabilistic theory of measurement uncertainty. A Bayesian primer: subjective probability and Bayes' theorem probability distributions (a concise reminder) Bayesian inference of continuous quantities Gaussian likelihood counting experiments bypassing Bayes' theorem for routine applications Bayesian unfolding. Further comments, examples and applications: miscellanea on general issues in probability and inference combination of experimental results - a closer look asymmetric uncertainties and nonlinear propagation which priors for frontier physics? Concluding matter: conclusions and bibliography.

Bayesian clustering and product partition models

Abstract: Summary. We present a decision theoretic formulation of product partition models (PPMs) that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously. A key observation in our construction is the fact that PPMs can be formulated in the context of model selection. The underlying partition structure in these models is closely related to that arising in connection with Dirichlet processes. This allows a straightforward adaptation of some computational strategies--originally devised for nonparametric Bayesian problems-to our framework. The resulting algorithms are more flexible than other competing alternatives that are used for problems involving PPMs. We propose an algorithm that yields Bayes estimates of the quantities of interest and the groups of experimental units. We explore the application of our methods to the detection of outliers in normal and Student t regression models, with clustering structure equivalent to that induced by a Dirichlet process prior. We also discuss the sensitivity of the results considering different prior distributions for the partitions.

Maximum Likelihood Estimation of Latent Affine Processes

Abstract: This article develops a direct filtration-based maximum likelihood methodology for estimating the parameters and realizations of latent affine processes. The equivalent of Bayes' rule is derived for recursively updating the joint characteristic function of latent variables and the data conditional upon past data. Likelihood functions can consequently be evaluated directly by Fourier inversion. An application to daily stock returns over 1953-96 reveals substantial divergences from EMM-based estimates: in particular, more substantial and time-varying jump risk.

A Bayesian semiparametric joint hierarchical model for longitudinal and survival data

Abstract: This article proposes a new semiparametric Bayesian hierarchical model for the joint modeling of longitudinal and survival data. We relax the distributional assumptions for the longitudinal model using Dirichlet process priors on the parameters defining the longitudinal model. The resulting posterior distribution of the longitudinal parameters is free of parametric constraints, resulting in more robust estimates. This type of approach is becoming increasingly essential in many applications, such as HIV and cancer vaccine trials, where patients' responses are highly diverse and may not be easily modeled with known distributions. An example will be presented from a clinical trial of a cancer vaccine where the survival outcome is time to recurrence of a tumor. Immunologic measures believed to be predictive of tumor recurrence were taken repeatedly during follow-up. We will present an analysis of this data using our new semiparametric Bayesian hierarchical joint modeling methodology to determine the association of these longitudinal immunologic measures with time to tumor recurrence.

Subjective and objective Bayesian statistics : principles, models, and applications

Abstract: Preface. Preface to the First Edition. A Bayesian Hall of Fame. PART I: FOUNDATIONS AND PRINCIPLES. 1. Background. 2. A Bayesian Perspective on Probability. 3. The Likelihood Function. 4. Bayes' Theorem. 5. Prior Distributions. PART II: NUMERICAL IMPLEMENTATION OF THE BAYESIAN PARADIGM. 6. Markov Chain Monte Carlo Methods (Siddhartha Chib). 7. Large Sample Posterior Distributions and Approximations. PART III: BAYESIAN STATISTICAL INFERENCE AND DECISION MAKING. 8. Bayesian Estimation. 9. Bayesian Hypothesis Testing. 10. Predictivism. 11. Bayesian Decision Making. PART IV: MODELS AND APPLICATIONS. 12. Bayesian Inference in the General Linear Model. 13. Model Averaging (Merlise Clyde). 14. Hierarchical Bayesian Modeling (Alan Zaslavsky). 15. Bayesian Factor Analysis. 16. Bayesian Inference in Classification and Discrimination. Description of Appendices. Appendix 1. Bayes, Thomas, (Hilary L. Seal). Appendix 2. Thomas Bayes. A Bibliographical Note (George A. Barnard). Appendix 3. Communication of Bayes' Essay to the Philosophical Transactions of the Royal Society of London (Richard Price). Appendix 4. An Essay Towards Solving a Problem in the Doctrine of Chances (Reverend Thomas Bayes). Appendix 5. Applications of Bayesian Statistical Science. Appendix 6. Selecting the Bayesian Hall of Fame. Appendix 7. Solutions to Selected Exercises. Bibliography. Subject Index. Author Index.

Optimal Search for a Lost Target in a Bayesian World.

Abstract: This paper presents a Bayesian approach to the problem of searching for a single lost target by a single autonomous sensor platform. The target may be static or mobile but not evading. Two candidate utility functions for the control solution are highlighted, namely the Mean Time to Detection, and the Cumulative Probability of Detection. The framework is implemented for an airborne vehicle looking for both a stationary and a drifting target at sea. Simulation results for different control solutions are investigated and compared to demonstrate the effectiveness of the method.

Probabilistic sensitivity analysis for decision trees with multiple branches: use of the Dirichlet distribution in a Bayesian framework.

Abstract: In structuring decision models of medical interventions, it is commonly recommended that only 2 branches be used for each chance node to avoid logical inconsistencies that can arise during sensitivity analyses if the branching probabilities do not sum to 1. However, information may be naturally available in an unconditional form, and structuring a tree in conditional form may complicate rather than simplify the sensitivity analysis of the unconditional probabilities. Current guidance emphasizes using probabilistic sensitivity analysis, and a method is required to provide probabilistic probabilities over multiple branches that appropriately represents uncertainty while satisfying the requirement that mutually exclusive event probabilities should sum to 1. The authors argue that the Dirichlet distribution, the multivariate equivalent of the beta distribution, is appropriate for this purpose and illustrate its use for generating a fully probabilistic transition matrix for a Markov model. Furthermore, they demonstrate that by adopting a Bayesian approach, the problem of observing zero counts for transitions of interest can be overcome.

Dynamic Bayesian Network and Nonparametric Regression for Nonlinear Modeling of Gene Networks from Time Series Gene Expression Data

Abstract: We propose a dynamic Bayesian network and nonparametric regression model for constructing a gene network from time series microarray gene expression data. The proposed method can overcome a shortcoming of the Bayesian network model in the sense of the construction of cyclic regulations. The proposed method can analyze the microarray data as continuous data and can capture even nonlinear relations among genes. It can be expected that this model will give a deeper insight into the complicated biological systems. We also derive a new criterion for evaluating an estimated network from Bayes approach. We demonstrate the effectiveness of our method by analyzing Saccharomyces cerevisiae gene expression data.

Bayesian network and nonparametric heteroscedastic regression for nonlinear modeling of genetic network

Abstract: We propose a new statistical method for constructing a genetic network from microarray gene expression data by using a Bayesian network. An essential point of Bayesian network construction is the estimation of the conditional distribution of each random variable. We consider fitting nonparametric regression models with heterogeneous error variances to the microarray gene expression data to capture the nonlinear structures between genes. Selecting the optimal graph, which gives the best representation of the system among genes, is still a problem to be solved. We theoretically derive a new graph selection criterion from Bayes approach in general situations. The proposed method includes previous methods based on Bayesian networks. We demonstrate the effectiveness of the proposed method through the analysis of Saccharomyces cerevisiae gene expression data newly obtained by disrupting 100 genes.

Stochastic Search Variable Selection for Identifying Multiple Quantitative Trait Loci

Abstract: In this article, we utilize stochastic search variable selection methodology to develop a Bayesian method for identifying multiple quantitative trait loci (QTL) for complex traits in experimental designs. The proposed procedure entails embedding multiple regression in a hierarchical normal mixture model, where latent indicators for all markers are used to identify the multiple markers. The markers with significant effects can be identified as those with higher posterior probability included in the model. A simple and easy-to-use Gibbs sampler is employed to generate samples from the joint posterior distribution of all unknowns including the latent indicators, genetic effects for all markers, and other model parameters. The proposed method was evaluated using simulated data and illustrated using a real data set. The results demonstrate that the proposed method works well under typical situations of most QTL studies in terms of number of markers and marker density.

Estimating Expected Information Gains for Experimental Designs With Application to the Random Fatigue-Limit Model

Abstract: Expected gain in Shannon information is commonly suggested as a Bayesian design evaluation criterion. Because estimating expected information gains is computationally expensive, examples in which they have been successfully used in identifying Bayes optimal designs are both few and typically quite simplistic. This article discusses in general some properties of estimators of expected information gains based on Markov chain Monte Carlo (MCMC) and Laplacian approximations. We then investigate some issues that arise when applying these methods to the problem of experimental design in the (technically nontrivial) random fatigue-limit model of Pascual and Meeker. An example comparing follow-up designs for a laminate panel study is provided.

Hypothesis testing and theory evaluation at the boundaries: Surprising insights from Bayes's theorem.

Abstract: Because the probability of obtaining an experimental finding given that the null hypothesis is true [p(F\H0)] is not the same as the probability that the null hypothesis is true given a finding [p(H0\F)], calculating the former probability does not justify conclusions about the latter one. As the standard null-hypothesis significance-testing procedure does just that, it is logically invalid (J. Cohen, 1994). Theoretically, Bayes's theorem yields p(H0\F), but in practice, researchers rarely know the correct values for 2 of the variables in the theorem. Nevertheless, by considering a wide range of possible values for the unknown variables, it is possible to calculate a range of theoretical values for p(H0\F) and to draw conclusions about both hypothesis testing and theory evaluation.