## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

Multiple testing on dependent count data faces two basic modelling elements: the choice of distributions under the null and the non-null states and the modelling of the dependence structure across observations. A Bayesian hidden Markov model is constructed for Poisson count data to handle these two issues. The proposed Bayesian method is based on the posterior probability of the null state and exhibits the property of an optimal test procedure, which has the lowest false-negative rate with the false discovery rate under control. Furthermore, the model has either single or mixture of Poisson distributions used under the non-null state. Model selection methods are employed here to decide the number of components in the mixture. Different approaches of calculating marginal likelihood are discussed. Extensive simulation studies and a case study are employed to examine and compare a collection of commonly used testing procedures and model selection criteria.

To read the full-text of this research,

you can request a copy directly from the authors.

... Different approaches have equally been explored in estimating parameters in the models, such as likelihood and quasi-likelihood. Some of these models are not robust in fitting count data that are over-dispersed and at the same time under-dispersed, such as the studies conducted by [20][21][22][23] . ...

The need to model count data correctly calls for the introduction of a flexible yet a strong model that can sufficiently handle various types of count data. Models such as Ordinary Least Squares (OLS) used in the past were considered unsuitable, and the introduction of the Generalized Linear Model (GLM) and its various extensions was the first breakthrough recorded in modelling count data. In this article, Bayesian Dirichlet process mixture prior of generalized linear mixed models (DPMglmm) was proposed. Metropolis Hasting Monte Carlo Markov Chain (M-H MCMC) was used to draw parameters from target posterior distribution. The Iterated Weighted Least Square (IWLS) proposal was used to determine the acceptance probability in the M-H MCMC phase. Under and over-dispersed count data were simulated, 500 Burn-in was scanned so as to allow for stability in the chain. 100 thinning interval was allowed so as to nullify the possible effect of autocorrelation in the data due to the Monte Carlo procedure. The DPMglmm and other competing models were fitted to the simulated data and real-life data sets of health insurance claims. The results obtained showed that DPMglmm outperformed MCMCglmm, Bayesian Discrete Weibull and four other frequentist models. This shows that DPMglmm is flexible, and can fit count data better, either under-dispersed or over-dispersed data.

Large-scale multiple testing with correlated tests and auxiliary statistics arises in a wide range of scientific fields. Conventional multiple testing procedures largely ignored auxiliary information, such as sparsity information, and the dependence structure among tests. This may result in loss of testing efficiency. In this paper, we propose a procedure, called multivariate local index of significance (mvLIS) procedure, for large-scale multiple testing. The mvLIS procedure can not only characterize local correlations among tests via a Markov chain but also incorporates auxiliary information via multivariate statistics. We present that the oracle mvLIS procedure is valid, namely, it controls false discovery rate (FDR) at the pre-specified level, and show that it yields the smallest false non-discovery rate (FNR) at the same FDR level. Then a data-driven mvLIS procedure is developed to mimic the oracle procedure. Comprehensive simulation studies and a real data analysis of schizophrenia (SCZ) data are performed to illustrate the superior performance of the mvLIS procedure. Moreover, as a byproduct that is of independent interest, we generalize the single-index modulated (SIM) multiple testing procedure, which embeds prior information via 2-dimensional p-values, to allow for d-dimensional (d≥3) statistics in multiple testing. The detailed extension is deferred to Discussion.

The traditional Poisson regression model for fitting count data is considered inadequate to fit over-or under-dispersed count data and new models have been developed to make up for such inadequacies inherent in the model. In this study, Bayesian Multi-level model was proposed using the No-U-Turn Sampler (NUTS) sampler to sample from the posterior distribution. A simulation was carried out for both over-and under-dispersed data from discrete Weibull distribution. Pareto k diagnostics was implemented, and the result showed that under-dispersed and over-dispersed simulated data has all its k value to be less than 0.5, which indicate that all the observations are good. Also all WAIC were the same as LOO-IC except for Poisson in the over-dispersed simulated data. Real-life data set from National Health Insurance Scheme (NHIS) was used for further analysis. Seven multi-level models were fitted and the Geometric model outperformed other model.

The probability of false discovery proportion (FDP) exceeding
$\gamma\in[0,1)$, defined as $\gamma$-FDP, has received much attention as a
measure of false discoveries in multiple testing. Although this measure has
received acceptance due to its relevance under dependency, not much progress
has been made yet advancing its theory under such dependency in a nonasymptotic
setting, which motivates our research in this article. We provide a larger
class of procedures containing the stepup analog of, and hence more powerful
than, the stepdown procedure in Lehmann and Romano [Ann. Statist. 33 (2005)
1138-1154] controlling the $\gamma$-FDP under similar positive dependence
condition assumed in that paper. We offer better alternatives of the stepdown
and stepup procedures in Romano and Shaikh [IMS Lecture Notes Monogr. Ser. 49
(2006a) 33-50, Ann. Statist. 34 (2006b) 1850-1873] using pairwise joint
distributions of the null $p$-values. We generalize the notion of $\gamma$-FDP
making it appropriate in situations where one is willing to tolerate a few
false rejections or, due to high dependency, some false rejections are
inevitable, and provide methods that control this generalized $\gamma$-FDP in
two different scenarios: (i) only the marginal $p$-values are available and
(ii) the marginal $p$-values as well as the common pairwise joint distributions
of the null $p$-values are available, and assuming both positive dependence and
arbitrary dependence conditions on the $p$-values in each scenario. Our
theoretical findings are being supported through numerical studies.

Traditional voxel-level multiple testing procedures in neuroimaging, mostly p-value based, often ignore the spatial correlations among neighboring voxels and thus suffer from substantial loss of power. We extend the local-significance-index based procedure originally developed for the hidden Markov chain models, which aims to minimize the false nondiscovery rate subject to a constraint on the false discovery rate, to three-dimensional neuroimaging data using a hidden Markov random field model. A generalized expectation-maximization algorithm for maximizing the penalized likelihood is proposed for estimating the model parameters. Extensive simulations show that the proposed approach is more powerful than conventional false discovery rate procedures. We apply the method to the comparison between mild cognitive impairment, a disease status with increased risk of developing Alzheimer's or another dementia, and normal controls in the FDG-PET imaging study of the Alzheimer's Disease Neuroimaging Initiative.
© 2015, The International Biometric Society.

Model determination is divided into the issues of model adequacy and model selection. Predictive distributions are used to address both issues. This seems natural since, typically, prediction is a primary purpose for the chosen model. A cross-validation viewpoint is argued for. In particular, for a given model, it is proposed to validate conditional predictive distributions arising from single point deletion against observed responses. Sampling based methods are used to carry out required calculations. An example investigates the adequacy of and rather subtle choice between two sigmoidal growth models of the same dimension.

A vast literature in statistics, biometrics, and econometrics is concerned with the analysis of binary and polychotomous response data. The classical approach fits a categorical response regression model using maximum likelihood, and inferences about the model are based on the associated asymptotic theory. The accuracy of classical confidence statements is questionable for small sample sizes. In this article, exact Bayesian methods for modeling categorical response data are developed using the idea of data augmentation. The general approach can be summarized as follows. The probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data. Values of the latent data can be simulated from suitable truncated normal distributions. If the latent data are known, then the posterior distribution of the parameters can be computed using standard results for normal linear models. Draws from this posterior are used to sample new latent data, and the process is iterated with Gibbs sampling. This data augmentation approach provides a general framework for analyzing binary regression models. It leads to the same simplification achieved earlier for censored regression models. Under the proposed framework, the class of probit regression models can be enlarged by using mixtures of normal distributions to model the latent data. In this normal mixture class, one can investigate the sensitivity of the parameter estimates to the choice of “link function,” which relates the linear regression estimate to the fitted probabilities. In addition, this approach allows one to easily fit Bayesian hierarchical models. One specific model considered here reflects the belief that the vector of regression coefficients lies on a smaller dimension linear subspace. The methods can also be generalized to multinomial response models with J > 2 categories. In the ordered multinomial model, the J categories are ordered and a model is written linking the cumulative response probabilities with the linear regression structure. In the unordered multinomial model, the latent variables have a multivariate normal distribution with unknown variance-covariance matrix. For both multinomial models, the data augmentation method combined with Gibbs sampling is outlined. This approach is especially attractive for the multivariate probit model, where calculating the likelihood can be difficult.

We introduce the weighted likelihood bootstrap (WLB) as a way to simulate approximately from a posterior distribution. This method is often easy to implement, requiring only an algorithm for calculating the maximum likelihood estimator, such as iteratively reweighted least squares. In the generic weighting scheme, the WLB is first order correct under quite general conditions. Inaccuracies can be removed by using the WLB as a source of samples in the sampling-importance resampling (SIR) algorithm, which also allows incorporation of particular prior information. The SIR- adjusted WLB can be a competitive alternative to other integration methods in certain models. Asymptotic expansions elucidate the second- order properties of the WLB, which is a generalization of Rubin’s Bayesian bootstrap [D. B. Rubin, Ann. Stat. 9, 130-134 (1981)]. The calculation of approximate Bayes factors for model comparison is also considered. We note that, given a sample simulated from the posterior distribution, the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values; a modification of this estimator that avoids instability is also noted. These methods provide simple ways of calculating approximate Bayes factors and posterior model probabilities for a very wide class of models.

Multiview video has recently emerged as a means to improve user experience in
novel multimedia services. We propose a new stochastic model to characterize
the traffic generated by a Multiview Video Coding (MVC) variable bit rate
source. To this aim, we resort to a Poisson Hidden Markov Model (P-HMM), in
which the first (hidden) layer represents the evolution of the video activity
and the second layer represents the frame sizes of the multiple encoded views.
We propose a method for estimating the model parameters in long MVC sequences.
We then present extensive numerical simulations assessing the model's ability
to produce traffic with realistic characteristics for a general class of MVC
sequences. We then extend our framework to network applications where we show
that our model is able to accurately describe the sender and receiver buffers
behavior in MVC transmission. Finally, we derive a model of user behavior for
interactive view selection, which, in conjunction with our traffic model, is
able to accurately predict actual network load in interactive multiview
services.

Large-scale multiple testing tasks often exhibit dependence, and leveraging
the dependence between individual tests is still one challenging and important
problem in statistics. With recent advances in graphical models, it is feasible
to use them to perform multiple testing under dependence. We propose a multiple
testing procedure which is based on a Markov-random-field-coupled mixture
model. The ground truth of hypotheses is represented by a latent binary Markov
random field, and the observed test statistics appear as the coupled mixture
variables. The parameters in our model can be automatically learned by a novel
EM algorithm. We use an MCMC algorithm to infer the posterior probability that
each hypothesis is null (termed local index of significance), and the false
discovery rate can be controlled accordingly. Simulations show that the
numerical performance of multiple testing can be improved substantially by
using our procedure. We apply the procedure to a real-world genome-wide
association study on breast cancer, and we identify several SNPs with strong
association evidence.

The objective of this paper is to quantify the effect of correlation in false discovery rate analysis. Specifically, we derive approximations for the mean, variance, distribution and quantiles of the standard false discovery rate estimator for arbitrarily correlated data. This is achieved using a negative binomial model for the number of false discoveries, where the parameters are found empirically from the data. We show that correlation may increase the bias and variance of the estimator substantially with respect to the independent case, and that in some cases, such as an exchangeable correlation structure, the estimator fails to be consistent as the number of tests becomes large.

The common approach to the multiplicity problem calls for controlling the familywise error rate (FWER). This approach, though, has faults, and we point out a few. A different approach to problems of multiple significance testing is presented. It calls for controlling the expected proportion of falsely rejected hypotheses – the false discovery rate. This error rate is equivalent to the FWER when all hypotheses are true but is smaller otherwise. Therefore, in problems where the control of the false discovery rate rather than that of the FWER is desired, there is potential for a gain in power. A simple sequential Bonferroni-type procedure is proved to control the false discovery rate for independent test statistics, and a simulation study shows that the gain in power is substantial. The use of the new procedure and the appropriateness of the criterion are illustrated with examples.

Genome-wide association studies (GWAS) are increasingly utilized for identifying novel susceptible genetic variants for complex traits, but there is little consensus on analysis methods for such data. Most commonly used methods include single single nucleotide polymorphism (SNP) analysis or haplotype analysis with Bonferroni correction for multiple comparisons. Since the SNPs in typical GWAS are often in linkage disequilibrium (LD), at least locally, Bonferroni correction of multiple comparisons often leads to conservative error control and therefore lower statistical power. In this paper, we propose a hidden Markov random field model (HMRF) for GWAS analysis based on a weighted LD graph built from the prior LD information among the SNPs and an efficient iterative conditional mode algorithm for estimating the model parameters. This model effectively utilizes the LD information in calculating the posterior probability that an SNP is associated with the disease. These posterior probabilities can then be used to define a false discovery controlling procedure in order to select the disease-associated SNPs. Simulation studies demonstrated the potential gain in power over single SNP analysis. The proposed method is especially effective in identifying SNPs with borderline significance at the single-marker level that nonetheless are in high LD with significant SNPs. In addition, by simultaneously considering the SNPs in LD, the proposed method can also help to reduce the number of false identifications of disease-associated SNPs. We demonstrate the application of the proposed HMRF model using data from a case-control GWAS of neuroblastoma and identify 1 new SNP that is potentially associated with neuroblastoma.

Motivation:
Genome-wide association studies (GWAS) interrogate common genetic variation across the entire human genome in an unbiased manner and hold promise in identifying genetic variants with moderate or weak effect sizes. However, conventional testing procedures, which are mostly P-value based, ignore the dependency and therefore suffer from loss of efficiency. The goal of this article is to exploit the dependency information among adjacent single nucleotide polymorphisms (SNPs) to improve the screening efficiency in GWAS.
Results:
We propose to model the linear block dependency in the SNP data using hidden Markov models (HMMs). A compound decision-theoretic framework for testing HMM-dependent hypotheses is developed. We propose a powerful data-driven procedure [pooled local index of significance (PLIS)] that controls the false discovery rate (FDR) at the nominal level. PLIS is shown to be optimal in the sense that it has the smallest false negative rate (FNR) among all valid FDR procedures. By re-ranking significance for all SNPs with dependency considered, PLIS gains higher power than conventional P-value based methods. Simulation results demonstrate that PLIS dominates conventional FDR procedures in detecting disease-associated SNPs. Our method is applied to analysis of the SNP data from a GWAS of type 1 diabetes. Compared with the Benjamini-Hochberg (BH) procedure, PLIS yields more accurate results and has better reproducibility of findings.
Conclusion:
The genomic rankings based on our procedure are substantially different from the rankings based on the P-values. By integrating information from adjacent locations, the PLIS rankings benefit from the increased signal-to-noise ratio, hence our procedure often has higher statistical power and better reproducibility. It provides a promising direction in large-scale GWAS.
Availability:
An R package PLIS has been developed to implement the PLIS procedure. Source codes are available upon request and will be available on CRAN (http://cran.r-project.org/).
Contact:
zhiwei@njit.edu
Supplementary information:
Supplementary data are available at Bioinformatics online.

Abstract The problem,of multiple testing for the presence of signal in spatial data can involve a large number of locations. Traditionally, each location is tested separately for signal presence but then the findings are reported in terms of clusters of nearby,locations. This is an indication that the units of interests for testing are clusters rather than individual locations. The investigator may know,a-priori these more,natural units or an approximation,to them.,We suggest testing these cluster units rather than individual locations, thus in- creasing the signal to noise ratio within the unit tested as well as reducing the number,of hypotheses,tests conducted.,Since the signal may be absent from part of each cluster, we define a cluster as containing signal if the sig- nal is present somewhere,within the cluster. We suggest controlling the false discovery rate (FDR) on clusters, i.e. the expected proportion of clusters rejected erroneously out of all clusters rejected, or its extension to general weights,(WFDR). We introduce,a powerful two-stage testing procedure,and show that it controls the WFDR. Once the cluster discoveries have been made, we suggest ’cleaning’ locations in which the signal is absent. For this purpose we develop a hierarchical testing procedure that tests clusters first, then loca- tions within rejected clusters. We show,formally that this procedure controls the desired location error rate asymptotically, and conjecture that this is so also for realistic settings by extensive simulations. We discuss an application to functional neuroimaging,which,motivated,this research and demonstrate the advantages,of the proposed,methodology,on an example. Key words : Signal detection, FDR, multiple testing, hierarchical testing, weighted 2

Some effort has been undertaken over the last decade to provide conditions for the control of the false discovery rate by the linear step-up procedure (LSU) for testing $n$ hypotheses when test statistics are dependent. In this paper we investigate the expected error rate (EER) and the false discovery rate (FDR) in some extreme parameter configurations when $n$ tends to infinity for test statistics being exchangeable under null hypotheses. All results are derived in terms of $p$-values. In a general setup we present a series of results concerning the interrelation of Simes' rejection curve and the (limiting) empirical distribution function of the $p$-values. Main objects under investigation are largest (limiting) crossing points between these functions, which play a key role in deriving explicit formulas for EER and FDR. As specific examples we investigate equi-correlated normal and $t$-variables in more detail and compute the limiting EER and FDR theoretically and numerically. A surprising limit behavior occurs if these models tend to independence.

Model determination is a fundamental data analytic task. Here we consider the problem of choosing among a finite (without loss of generality we assume two) set of models. After briefly reviewing classical and Bayesian model choice strategies we present a general predictive density which includes all proposed Bayesian approaches that we are aware of. Using Laplace approximations we can conveniently assess and compare the asymptotic behaviour of these approaches. Concern regarding the accuracy of these approximations for small to moderate sample sizes encourages the use of Monte Carlo techniques to carry out exact calculations. A data set fitted with nested non‐linear models enables comparisons between proposals and between exact and asymptotic values.

An optimal and flexible multiple hypotheses testing procedure is constructed for dependent data based on Bayesian techniques, aiming at handling two challenges, namely dependence structure and non-null distribution specification. Ignoring dependence among hypotheses tests may lead to loss of efficiency and bias in decision. Misspecification in the non-null distribution, on the other hand, can result in both false positive and false negative errors. Hidden Markov models are used to accommodate the dependence structure among the tests. Dirichlet mixture process prior is applied on the non-null distribution to overcome the potential pitfalls in distribution misspecification. The testing algorithm based on Bayesian techniques optimizes the false negative rate (FNR) while controlling the false discovery rate (FDR). The procedure is applied to pointwise and clusterwise analysis. Its performance is compared with existing approaches using both simulated and real data examples.

The Weather Risk Attribution Forecast (WRAF) is a forecasting tool that uses output from global climate models to make simultaneous attribution statements about whether and how greenhouse gas emissions have contributed to extreme weather across the globe. However, in conducting a large number of simultaneous hypothesis tests, the WRAF is prone to identifying false "discoveries." A common technique for addressing this multiple testing problem is to adjust the procedure in a way that controls the proportion of true null hypotheses that are incorrectly rejected, or the false discovery rate (FDR). Unfortunately, generic FDR procedures suffer from low power when the hypotheses are dependent, and techniques designed to account for dependence are sensitive to misspecification of the underlying statistical model. In this paper, we develop a Bayesian decision theoretic approach for dependent multiple testing that flexibly controls false discovery and is robust to model misspecification. We illustrate the robustness of our procedure to model error with a simulation study, using a framework that accounts for generic spatial dependence and allows the practitioner to flexibly specify the decision criteria. Finally, we outline the best procedure of those considered for use in the WRAF workflow and apply the procedure to several seasonal forecasts.

In a 1935 paper and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is one-half. Although there has been much discussion of Bayesian hypothesis testing in the context of criticism of P-values, less attention has been given to the Bayes factor as a practical tool of applied statistics. In this article we review and discuss the uses of Bayes factors in the context of five scientific applications in genetics, sports, ecology, sociology, and psychology. We emphasize the following points:

The paper develops a unified theoretical and computational framework for false discovery control in multiple testing of spatial signals. We consider both pointwise and clusterwise spatial analyses, and derive oracle procedures which optimally control the false discovery rate, false discovery exceedance and false cluster rate. A data-driven finite approximation strategy is developed to mimic the oracle procedures on a continuous spatial domain. Our multiple-testing procedures are asymptotically valid and can be effectively implemented using Bayesian computational algorithms for analysis of large spatial data sets. Numerical results show that the procedures proposed lead to more accurate error control and better power performance than conventional methods. We demonstrate our methods for analysing the time trends in tropospheric ozone in eastern USA.

RNA sequencing (RNA-Seq) allows for the identification of novel exon-exon junctions and quantification of gene expression levels. We show that from RNA-Seq data one may also detect utilization of alternative polyadenylation (APA) in 3' untranslated regions (3'UTRs) known to play a critical role in the regulation of mRNA stability, cellular localization and translation efficiency. Given the dynamic nature of APA, it is desirable to examine the APA on a sample by sample basis. We used a Poisson hidden Markov model (PHMM) of RNA-Seq data to identify potential APA in human liver and brain cortex tissues leading to shortened 3' UTRs. Over three hundred transcripts with shortened 3'UTRs were detected with sensitivity >75% and specificity >60%. Tissue-specific 3'UTR shortening was observed for 32 genes with a q-value≤0.1. When compared to alternative isoforms detected by Cufflinks or MISO, our PHMM method agreed on over 100 transcripts with shortened 3' UTRs. Given the increasing usage of RNA-Seq for gene expression profiling, using PHMM to investigate sample-specific 3'UTR shortening could be an added benefit from this emerging technology.

Let pi(w) ,i =1 , 2, be two densities with common support where each density is known up to a normalizing constant: pi(w )= qi(w)/ci .W e have draws from each density (e.g., via Markov chain Monte Carlo), and we want to use these draws to simulate the ratio of the normalizing constants, c1/c2. Such a compu- tational problem is often encountered in likelihood and Bayesian inference, and arises in fields such as physics and genetics. Many methods proposed in statistical and other literature (e.g., computational physics) for dealing with this problem are based on various special cases of the following simple identity: c1 c2 = E2(q1(w)α(w)) E1(q2(w)α(w)) . Here Ei denotes the expectation with respect to pi (i =1 , 2), and α is an arbitrary function such that the denominator is non-zero. A main purpose of this paper is to provide a theoretical study of the usefulness of this identity, with focus on (asymptotically) optimal and practical choices of α. Using a simple but informa- tive example, we demonstrate that with sensible (not necessarily optimal) choices of α, we can reduce the simulation error by orders of magnitude when compared to the conventional importance sampling method, which corresponds to α =1 /q2. We also introduce several generalizations of this identity for handling more compli- cated settings (e.g., estimating several ratios simultaneously) and pose several open problems that appear to have practical as well as theoretical value. Furthermore, we discuss related theoretical and empirical work.

In this paper we develop a very general Markov switching model for analysis of economic time series. Our general set-up allows us to assess the effects of a variety of model and prior assumptions on the results. The growth rates of US quarterly real gross national product are used to illustrate the proposed analysis. We find that although the evidence is not strong the analysis does not support the model in which the dynamic behavior is constant and that allowing the dynamic structure to change affects the results. We also find that the results are sensitive to the prior specification.

Time-series of count data are generated in many different contexts, such as web access logging, freeway traffic mon- itoring, and security logs associated with buildings. Since this data measures the aggregated behavior of individual human beings, it typically exhibits a periodicity in time on a number of scales (daily, weekly, etc.) that reflects the rhythms of the underlying human activity and makes the data appear non-homogeneous. At the same time, the data is often corrupted by a number of bursty periods of unusual behavior such as building events, traffic accidents, and so forth. The data mining problem of finding and extracting these anomalous events is made difficult by both of these elements. In this paper we describe a framework for unsu- pervised learning in this context, based on a time-varying Poisson process model that can also account for anomalous events. We show how the parameters of this model can be learned from count time series using statistical estima- tion techniques. We demonstrate the utility of this model on two data sets for which we have partial ground truth in the form of known events, one from freeway traffic data and another from building access data, and show that the model performs significantly better than a non-probabilistic, threshold-based technique. We also describe how the model can be used to investigate different degrees of periodicity in the data, including systematic day-of-week and time-of- day effects, and make inferences about the detected events (e.g., popularity or level of attendance). Our experimen- tal results indicate that the proposed time-varying Poisson model provides a robust and accurate framework for adap- tively and autonomously learning how to separate unusual bursty events from traces of normal human activity.

The paper considers the problem of multiple testing under dependence in a compound decision theoretic framework. The observed data are assumed to be generated from an underlying two-state hidden Markov model. We propose oracle and asymptotically optimal data-driven procedures that aim to minimize the false non-discovery rate FNR subject to a constraint on the false discovery rate FDR. It is shown that the performance of a multiple-testing procedure can be substantially improved by adaptively exploiting the dependence structure among hypotheses, and hence conventional FDR procedures that ignore this structural information are inefficient. Both theoretical properties and numerical performances of the procedures proposed are investigated. It is shown that the procedures proposed control FDR at the desired level, enjoy certain optimality properties and are especially powerful in identifying clustered non-null cases. The new procedure is applied to an influenza-like illness surveillance study for detecting the timing of epidemic periods. Copyright (c) 2009 Royal Statistical Society.

This paper surveys the fundamental principles of subjective Bayesian inference in econometrics and the implementation of those principles using posterior simulation methods. The emphasis is on the combination of models and the development of predictive distributions. Moving beyond conditioning on a fixed number of completely specified models, the paper introduces subjective Bayesian tools for formal comparison of these models with as yet incompletely specified models. The paper then shows how posterior simulators can facilitate communication between investigators (for example, econometricians) on the one hand and remote clients (for example, decision makers) on the other, enabling clients to vary the prior distributions and functions of interest employed by investigators. A theme of the paper is the practicality of subjective Bayesian methods. To this end, the paper describes publicly available software for Bayesian inference, model development, and communication and provides illustrations using two simple econometric models.

Hidden Markov models lead to intricate computational problems when considered directly. In this paper, we propose an approximation method based on Gibbs sampling which allows an effective derivation of Bayes estimators for these models.

This paper concerns the use and implementation of maximum-penalized-likelihood procedures for choosing the number of mixing components and estimating the parameters in independent and Markov-dependent mixture models. Computation of the estimates is achieved via algorithms for the automatic generation of starting values for the EM algorithm. Computation of the information matrix is also discussed. Poisson mixture models are applied to a sequence of counts of movements by a fetal lamb in utero obtained by ultrasound. The resulting estimates are seen to provide plausible mechanisms for the physiological process.

This paper discusses a model for a time series of epileptic seizure counts in which the mean of a Poisson distribution changes according to an underlying two-state Markov chain. The EM algorithm (Dempster, Laird, and Rubin, 1977, Journal of the Royal Statistical Society, Series B 39, 1-38) is used to compute maximum likelihood estimators for the parameters of this two-state mixture model and extensions are made allowing for nonstationarity. The model is illustrated using daily seizure counts for patients with intractable epilepsy and results are compared with a simple Poisson distribution and Poisson regressions. Some simulation results are also presented to demonstrate the feasibility of this model.

This paper discusses the problem of estimating marginal likelihoods for mixture and Markov switching model. Estimation is based on the method of bridge sampling (Meng and Wong 1996; Statistica Sinica 11, 552-86.) where Markov Chain Monte Carlo (MCMC) draws from the posterior density are combined with an i.i.d. sample from an importance density. The importance density is constructed in an unsupervised manner from the MCMC draws using a mixture of complete data posteriors. Whereas the importance sampling estimator as well as the reciprocal importance sampling estimator are sensitive to the tail behaviour of the importance density, we demonstrate that the bridge sampling estimator is far more robust. Our case studies range from computing marginal likelihoods for a mixture of multivariate normal distributions, testing for the inhomogeneity of a discrete time Poisson process, to testing for the presence of Markov switching and order selection in the MSAR model. Copyright Royal Economic Socciety 2004

In high throughput genomic work, a very large number "d" of hypotheses are tested based on "n">"d" data samples. The large number of tests necessitates an adjustment for false discoveries in which a true null hypothesis was rejected. The expected number of false discoveries is easy to obtain. Dependences between the hypothesis tests greatly affect the variance of the number of false discoveries. Assuming that the tests are independent gives an inadequate variance formula. The paper presents a variance formula that takes account of the correlations between test statistics. That formula involves "O"("d"-super-2) correlations, and so a naïve implementation has cost "O"("nd"-super-2). A method based on sampling pairs of tests allows the variance to be approximated at a cost that is independent of "d". Copyright 2005 Royal Statistical Society.

Multiple-hypothesis testing involves guarding against much more complicated errors than single-hypothesis testing. Whereas we typically control the type I error rate for a single-hypothesis test, a compound error rate is controlled for multiple-hypothesis tests. For example, controlling the false discovery rate FDR traditionally involves intricate sequential "p"-value rejection methods based on the observed data. Whereas a sequential "p"-value method fixes the error rate and "estimates" its corresponding rejection region, we propose the opposite approach-we "fix" the rejection region and then estimate its corresponding error rate. This new approach offers increased applicability, accuracy and power. We apply the methodology to both the positive false discovery rate pFDR and FDR, and provide evidence for its benefits. It is shown that pFDR is probably the quantity of interest over FDR. Also discussed is the calculation of the "q"-value, the pFDR analogue of the "p"-value, which eliminates the need to set the error rate beforehand as is traditionally done. Some simple numerical examples are presented that show that this new approach can yield an increase of over eight times in power compared with the Benjamini-Hochberg FDR method. Copyright 2002 Royal Statistical Society.

Large-scale hypothesis testing problems, with hundreds or thousands of test statis- tics "zi" to consider at once, have become familiar in current practice. Applications of popular analysis methods such as false discovery rate techniques do not require independence of the zi's, but their accuracy can be compromised in high-correlation situations. This paper presents computational and theoretical methods for assessing the size and effect of correlation in large-scale testing. A simple theory leads to the identification of a single omnibus measure of correlation for the zi's order statistic. The theory relates to the correct choice of a null distribution for simultaneous significance testing, and its effect on inference.

this paper, we propose some solutions to these problems. Our goal is to come up with a simple, practical method for estimating the density. This is an interesting problem in its own right, as well as a first step towards solving other inference problems, such as providing more flexible distributions in hierarchical models. To see why the posterior is improper under the usual reference prior, we write the model in the following way. Let Z = (Z 1 ; : : : ; Z n ) and X = (X 1 ; : : : ; X n ). The Z

- Orfanogiannaki K

Identifying seismicity levels via Poisson hidden Markov models. (report)

- K Orfanogiannaki
- D Karlis
- G A Papadopoulos