Discretizing a Normal Prior for Change Point Estimation in Switching Regressions

Bayes decision procedures are considered for change point estimation in the simple bilinear segmented model. A discretized normal prior density is employed as the prior distribution for the change point index. Posterior probability functions are developed for this index under a vague prior formulation on the regression parameters. The procedure is applied to an example involving mercury toxicity data.     

Testing a molecular clock without an outgroup: derivations of induced priors on branch-length restrictions in a Bayesian framework

We propose a Bayesian method for testing molecular clock hypotheses for use with aligned sequence data from multiple taxa. Our method utilizes a nonreversible nucleotide substitution model to avoid the necessity of specifying either a known tree relating the taxa or an outgroup for rooting the tree. We employ reversible jump Markov chain Monte Carlo to sample from the posterior distribution of the phylogenetic model parameters and conduct hypothesis testing using Bayes factors, the ratio of the posterior to prior odds of competing models. Here, the Bayes factors reflect the relative support of the sequence data for equal rates of evolutionary change between taxa versus unequal rates, averaged over all possible phylogenetic parameters, including the tree and root position. As the molecular clock model is a restriction of the more general unequal rates model, we use the Savage-Dickey ratio to estimate the Bayes factors. The Savage-Dickey ratio provides a convenient approach to calculating Bayes factors in favor of sharp hypotheses. Critical to calculating the Savage-Dickey ratio is a determination of the prior induced on the modeling restrictions. We demonstrate our method on a well-studied mtDNA sequence data set consisting of nine primates. We find strong support against a global molecular clock, but do find support for a local clock among the anthropoids. We provide mathematical derivations of the induced priors on branch length restrictions assuming equally likely trees. These derivations also have more general applicability to the examination of prior assumptions in Bayesian phylogenetics.     

Bayesian Estimation in a Generalized Negative Binomial Distribution

A generalized negative binomial (GNB) distribution was introduced by JAIN and CONSUL (1971) and was modified by NELSON (1975). The probability function of the distribution is defined by the function p(x; m, β, θ)= θ^x (1 - θ)^m+βx—x for x=0, 1, …, and zero otherwise, where m>0, 0<θ<1 and β=0 or 1≦β<θ^?1. The Bayes estimators for a number of parametric functions of θ when m and β are known are derived. The prior information on θ may be given by a beta distribution, B(a, b), to which no subjective significance is attached. It has been illustrated that the parameters in the prior distribution can be assigned by a computer. Comparisons are made of the Bayes estimate of P(X=k) to the corresponding ML estimate and the MVU estimate for any given sample to the order n^?1 for different values of k..     

Integrating biological knowledge into variable selection: an empirical Bayes approach with an application in cancer biology

ABSTRACT: BACKGROUND: An important question in the analysis of biochemical data is that of identifying subsets of molecular variables that may jointly influence a biological response. Statistical variable selection methods have been widely used for this purpose. In many settings, it may be important to incorporate ancillary biological information concerning the variables of interest. Pathway and network maps are one example of a source of such information. However, although ancillary information is increasingly available, it is not always clear how it should be used nor how it should be weighted in relation to primary data. RESULTS: We put forward an approach in which biological knowledge is incorporated using informative prior distributions over variable subsets, with prior information selected and weighted in an automated, objective manner using an empirical Bayes formulation. We employ continuous, linear models with interaction terms and exploit biochemically-motivated sparsity constraints to permit exact inference. We show an example of priors for pathway- and network-based information and illustrate our proposed method on both synthetic response data and by an application to cancer drug response data. Comparisons are also made to alternative Bayesian and frequentist penalised-likelihood methods for incorporating network-based information. CONCLUSIONS: The empirical Bayes method proposed here can aid prior elicitation for Bayesian variable selection studies and help to guard against mis-specification of priors. Empirical Bayes, together with the proposed pathway-based priors, results in an approach with a competitive variable selection performance. In addition, the overall procedure is fast, deterministic, and has very few user-set parameters, yet is capable of capturing interplay between molecular players. The approach presented is general and readily applicable in any setting with multiple sources of biological prior knowledge.     

A Bayes factor approach with informative prior for rare genetic variant analysis from next generation sequencing data

The discovery of rare genetic variants through next generation sequencing is a very challenging issue in the field of human genetics. We propose a novel region‐based statistical approach based on a Bayes Factor (BF) to assess evidence of association between a set of rare variants (RVs) located on the same genomic region and a disease outcome in the context of case‐control design. Marginal likelihoods are computed under the null and alternative hypotheses assuming a binomial distribution for the RV count in the region and a beta or mixture of Dirac and beta prior distribution for the probability of RV. We derive the theoretical null distribution of the BF under our prior setting and show that a Bayesian control of the false Discovery Rate can be obtained for genome‐wide inference. Informative priors are introduced using prior evidence of association from a Kolmogorov‐Smirnov test statistic. We use our simulation program, sim1000G, to generate RV data similar to the 1000 genomes sequencing project. Our simulation studies showed that the new BF statistic outperforms standard methods (SKAT, SKAT‐O, Burden test) in case‐control studies with moderate sample sizes and is equivalent to them under large sample size scenarios. Our real data application to a lung cancer case‐control study found enrichment for RVs in known and novel cancer genes. It also suggests that using the BF with informative prior improves the overall gene discovery compared to the BF with noninformative prior.     

Models for estimating bayes factors with applications to phylogeny and tests of monophyly

Bayes factors comparing two or more competing hypotheses are often estimated by constructing a Markov chain Monte Carlo (MCMC) sampler to explore the joint space of the hypotheses. To obtain efficient Bayes factor estimates, Carlin and Chib (1995, Journal of the Royal Statistical Society, Series B57, 473-484) suggest adjusting the prior odds of the competing hypotheses so that the posterior odds are approximately one, then estimating the Bayes factor by simple division. A byproduct is that one often produces several independent MCMC chains, only one of which is actually used for estimation. We extend this approach to incorporate output from multiple chains by proposing three statistical models. The first assumes independent sampler draws and models the hypothesis indicator function using logistic regression for various choices of the prior odds. The two more complex models relax the independence assumption by allowing for higher-lag dependence within the MCMC output. These models allow us to estimate the uncertainty in our Bayes factor calculation and to fully use several different MCMC chains even when the prior odds of the hypotheses vary from chain to chain. We apply these methods to calculate Bayes factors for tests of monophyly in two phylogenetic examples. The first example explores the relationship of an unknown pathogen to a set of known pathogens. Identification of the unknown's monophyletic relationship may affect antibiotic choice in a clinical setting. The second example focuses on HIV recombination detection. For potential clinical application, these types of analyses must be completed as efficiently as possible.     

Bayes estimates as an approximation to maximum likelihood estimates

Ronald A. Fisher, who is the founder of maximum likelihood estimation (ML estimation), criticized the Bayes estimation of using a uniform prior distribution, because we can create estimates arbitrarily if we use Bayes estimation by changing the transformation used before the analysis. Thus, the Bayes estimates lack the scientific objectivity, especially when the amount of data is small. However, we can use the Bayes estimates as an approximation to the objective ML estimates if we use an appropriate transformation that makes the posterior distribution close to a normal distribution. One-to-one correspondence exists between a uniform prior distribution under a transformed scale and a non-uniform prior distribution under the original scale. For this reason, the Bayes estimation of ML estimates is essentially identical to the estimation using Jeffreys prior.     

Empirical Bayes estimation for combinations of multivariate bioassays

This article presents a new empirical Bayes estimator (EBE) and a shrinkage estimator for determining the relative potency from several multivariate bioassays by incorporating prior information on the model parameters based on Jeffreys' rules. The EBE can account for any extra variability among the bioassays, and if this extra variability is 0, then the EBE reduces to the maximum likelihood estimator for combinations of multivariate bioassays. The shrinkage estimator turns out to be a compromise of the prior information and the estimator from each multivariate bioassay, with the weights depending on the prior variance.     

Sensitivity of Bayes estimators to hyper-parameters with an application to maximum yield from fisheries

Priors are seldom unequivocal and an important component of Bayesian modeling is assessment of the sensitivity of the posterior to the specified prior distribution. This is especially true in fisheries science where the Bayesian approach has been promoted as a rigorous method for including existing information from previous surveys and from related stocks or species. These informative priors may be highly contested by various interest groups. Here, formulae for the first and second derivatives of Bayes estimators with respect to hyper-parameters of the joint prior density are given. The formula for the second derivative provides a correction to a previously published result. The formulae are shown to reduce to very convenient and easily implemented forms when the hyper-parameters are for exponential family marginal priors. For model parameters with such priors it is shown that the ratio of posterior variance to prior variance can be interpreted as the sensitivity of the posterior mean to the prior mean. This methodology is applied to a nonlinear state-space model for the biomass of South Atlantic albacore tuna and sensitivity of the maximum sustainable yield to the prior specification is examined.     

Likelihood and Bayes estimation of ancestral population sizes in hominoids using data from multiple loci

Polymorphisms in an ancestral population can cause conflicts between gene trees and the species tree. Such conflicts can be used to estimate ancestral population sizes when data from multiple loci are available. In this article I extend previous work for estimating ancestral population sizes to analyze sequence data from three species under a finite-site nucleotide substitution model. Both maximum-likelihood (ML) and Bayes methods are implemented for joint estimation of the two speciation dates and the two population size parameters. Both methods account for uncertainties in the gene tree due to few informative sites at each locus and make an efficient use of information in the data. The Bayes algorithm using Markov chain Monte Carlo (MCMC) enjoys a computational advantage over ML and also provides a framework for incorporating prior information about the parameters. The methods are applied to a data set of 53 nuclear noncoding contigs from human, chimpanzee, and gorilla published by Chen and Li. Estimates of the effective population size for the common ancestor of humans and chimpanzees by both ML and Bayes methods are approximately 12,000-21,000, comparable to estimates for modern humans, and do not support the notion of a dramatic size reduction in early human populations. Estimates published previously from the same data are several times larger and appear to be biased due to methodological deficiency. The divergence between humans and chimpanzees is dated at approximately 5.2 million years ago and the gorilla divergence 1.1-1.7 million years earlier. The analysis suggests that typical data sets contain useful information about the ancestral population sizes and that it is advantageous to analyze data of several species simultaneously.     

Bayesian Reliability Estimation of a Two-parameter Cauchy Distribution

This paper gives an approximate Bayes procedure for the estimation of the reliability function of a two-parameter Cauchy distribution using Jeffreys' non-informative prior with a squared-error loss function, and with a log-odds ratio squared-error loss function. Based on a Monte Carlo simulation study, two such Bayes estimators of the reliability are compared with the maximum likelihood estimator.     

Bayesian Estimation Using Ranked Set Sampling

Bayesian Estimation of the parameter of a distribution is considered using Ranked set sampling (RSS). It is shown that for at least one RSS plan, the Bayes estimator has smaller Bayes risk than the Bayes estimator using simple random sampling (SRS). Furthermore, for exponential family with conjugate prior, the Bayes estimator of the mean using balanced RSS dominates, in terms of its Bayes risk, the Bayes estimator of the mean using SRS. This procedure is used to estimate the average Milk yield of four hundreds and two sheep. The empirical efficiency supports the theoretical findings.     

Bayes empirical bayes inference of amino acid sites under positive selection

Codon-based substitution models have been widely used to identify amino acid sites under positive selection in comparative analysis of protein-coding DNA sequences. The nonsynonymous-synonymous substitution rate ratio (d(N)/d(S), denoted omega) is used as a measure of selective pressure at the protein level, with omega > 1 indicating positive selection. Statistical distributions are used to model the variation in omega among sites, allowing a subset of sites to have omega > 1 while the rest of the sequence may be under purifying selection with omega < 1. An empirical Bayes (EB) approach is then used to calculate posterior probabilities that a site comes from the site class with omega > 1. Current implementations, however, use the naive EB (NEB) approach and fail to account for sampling errors in maximum likelihood estimates of model parameters, such as the proportions and omega ratios for the site classes. In small data sets lacking information, this approach may lead to unreliable posterior probability calculations. In this paper, we develop a Bayes empirical Bayes (BEB) approach to the problem, which assigns a prior to the model parameters and integrates over their uncertainties. We compare the new and old methods on real and simulated data sets. The results suggest that in small data sets the new BEB method does not generate false positives as did the old NEB approach, while in large data sets it retains the good power of the NEB approach for inferring positively selected sites.     

Bayes Estimator of Survival Probability from Randomly Censored Observations with Weibull Distribution of Time to Death

This paper deals with Bayes estimation of survival probability when the data are randomly censored. Such a situation arises in case of a clinical trial which extends for a limited period T. A fixed number of patients (n) are observed whose times to death have identical Weibull distribution with parameters β and θ. The maximum times of observation for different patients are also independent uniform variables as the patients arrive randomly throughout the trial. For the joint prior distribution of (β, θ) as suggested by Sinha and Kale (1980, page 137) Bayes estimator of survival probability at time t (0<t<T) has been obtained. Considering squared error loss function it is the mean of the survival probability with respect to the posterior distribution of (β, θ). This estimator is then compared with the maximum likelihood estimator, by simulation, for various values of β, θ and censoring percentage. The proposed estimator is found to be better under certain conditions.     

Bayesian Estimation of the Distribution Function of the Poisson Model

This paper presents the Bayes estimators of the Poisson distribution function based on complete and truncated data under a natural conjugate prior. Laplace transform of the incomplete gamma function and the Gauss hypergeometric function have been employed in order to overcome the intractability of the integrals. Numerical examples from biosciences are given to illustrate the results. A Monte Carlo study has been carried out to compare Bayes estimators under complete data with the corresponding maximum liklihood estimators.     

Testing and estimating the non-disjunction fraction in Meiosis I using reference priors

In this paper we analyze the fraction of non-disjunction in Meiosis I assuming reference (non-informative) priors. We consider Jeffreys's approach to built a non-informative prior (Jeffreys's prior) for the fraction of non-disjunction in Meiosis I. We prove that Jeffreys's prior is a proper distribution. We perform Monte Carlo studies in order to compare Bayes estimates obtained assuming Jeffreys's and uniform priors. We consider full Bayesian significance test (FBST) and Bayes factor (BF) for testing precise hypothesis on the fraction of non-disjunction in Meiosis I. The ultimate goal of this paper is to compare these two test procedures through simulation studies using both prior specifications. An application to Down Syndrome data is also presented.     

Applications of the Dirichlet distribution to forensic match probabilities

The Dirichlet distribution provides a convenient conjugate prior for Bayesian analyses involving multinomial proportions. In particular, allele frequency estimation can be carried out with a Dirichlet prior. If data from several distinct populations are available, then the parameters characterizing the Dirichlet prior can be estimated by maximum likelihood and then used for allele frequency estimation in each of the separate populations. This empirical Bayes procedure tends to moderate extreme multinomial estimates based on sample proportions. The Dirichlet distribution can also be employed to model the contributions from different ancestral populations in computing forensic match probabilities. If the ancestral populations are in genetic equilibrium, then the product rule for computing match probabilities is valid conditional on the ancestral contributions to a typical person of the reference population. This fact facilitates computation of match probabilities and tight upper bounds to match probabilities.Editor's commentsThe author continues the formal Bayesian analysis introduced by Gjertson & Morris in this voluem. He invokes Dirichlet distributions, and so brings rigor to the discussion of the effects of population structure on match probabilities. The increased computational burden this approach entails should not be regarded as a hindrance.     

How Bayes tests of molecular phylogenies compare with frequentist approaches

MOTIVATION: The desire to compare molecular phylogenies has stimulated the design of numerous tests. Most of these tests are formulated in a frequentist framework, and it is not known how they compare with Bayes procedures. I propose here two new Bayes tests that either compare pairs of trees (Bayes hypothesis test, BHT), or test each tree against an average of the trees included in the analysis (Bayes significance test, BST). RESULTS: The algorithm, based on a standard Metropolis-Hastings sampler, integrates nuisance parameters out and estimates the probability of the data under each topology. These quantities are used to estimate Bayes factors for composite vs. composite hypotheses. Based on two data sets, the BHT and BST are shown to construct similar confidence sets to the bootstrap and the Shimodaira Hasegawa test, respectively. This suggests that the known difference among previous tests is mainly due to the null hypothesis considered.     

Modulations of EEG Beta Power during Planning and Execution of Grasping Movements

Although beta oscillations (≈ 13–35 Hz) are often considered as a sensorimotor rhythm, their functional role remains debated. In particular, the modulations of beta power during preparation and execution of complex movements in different contexts were barely investigated. Here, we analysed the beta oscillations recorded with electroencephalography (EEG) in a precued grasping task in which we manipulated two critical parameters: the grip type (precision vs. side grip) and the force (high vs. low force) required to pull an object along a horizontal axis. A cue was presented 3 s before a GO signal and provided full, partial or no information about the two movement parameters. We measured beta power over the centro-parietal areas during movement preparation and execution as well as during object hold. We explored the modulations of power in relation to the amount and type of prior information provided by the cue. We also investigated how beta power was affected by the grip and force parameters.We observed an increase in beta power around the cue onset followed by a decrease during movement preparation and execution. These modulations were followed by a transient power increase during object hold. This pattern of modulations did not differ between the 4 movement types (2 grips ×2 forces). However, the amount and type of prior information provided by the cue had a significant effect on the beta power during the preparatory delay. We discuss how these results fit with current hypotheses on the functional role of beta oscillations.     

Empirical and hierarchical Bayesian estimation of ancestral states

Several methods have been proposed to infer the states at the ancestral nodes on a phylogeny. These methods assume a specific tree and set of branch lengths when estimating the ancestral character state. Inferences of the ancestral states, then, are conditioned on the tree and branch lengths being true. We develop a hierarchical Bayes method for inferring the ancestral states on a tree. The method integrates over uncertainty in the tree, branch lengths, and substitution model parameters by using Markov chain Monte Carlo. We compare the hierarchical Bayes inferences of ancestral states with inferences of ancestral states made under the assumption that a specific tree is correct. We find that the methods are correlated, but that accommodating uncertainty in parameters of the phylogenetic model can make inferences of ancestral states even more uncertain than they would be in an empirical Bayes analysis.