Bayesian phylogenetic model selection using reversible jump Markov chain Monte Carlo

A common problem in molecular phylogenetics is choosing a model of DNA substitution that does a good job of explaining the DNA sequence alignment without introducing superfluous parameters. A number of methods have been used to choose among a small set of candidate substitution models, such as the likelihood ratio test, the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and Bayes factors. Current implementations of any of these criteria suffer from the limitation that only a small set of models are examined, or that the test does not allow easy comparison of non-nested models. In this article, we expand the pool of candidate substitution models to include all possible time-reversible models. This set includes seven models that have already been described. We show how Bayes factors can be calculated for these models using reversible jump Markov chain Monte Carlo, and apply the method to 16 DNA sequence alignments. For each data set, we compare the model with the best Bayes factor to the best models chosen using AIC and BIC. We find that the best model under any of these criteria is not necessarily the most complicated one; models with an intermediate number of substitution types typically do best. Moreover, almost all of the models that are chosen as best do not constrain a transition rate to be the same as a transversion rate, suggesting that it is the transition/transversion rate bias that plays the largest role in determining which models are selected. Importantly, the reversible jump Markov chain Monte Carlo algorithm described here allows estimation of phylogeny (and other phylogenetic model parameters) to be performed while accounting for uncertainty in the model of DNA substitution.     

Constant Hazard Function Models with a Change Point: A Bayesian Analysis Using Markov Chain Monte Carlo Methods

Metropolis algorithms along with Gibbs steps are proposed to perform a Bayesian analysis for change-point constant hazard function models considering different prior densities for the parameters and censored survival data. We also present some generalizations for the comparison of two treatments. The methodology is illustrated with some examples.     

Modeling kinetics of a large‐scale fed‐batch CHO cell culture by Markov chain Monte Carlo method

Markov chain Monte Carlo (MCMC) method was applied to model kinetics of a fed‐batch Chinese hamster ovary cell culture process in 5,000‐L bioreactors. The kinetic model consists of six differential equations, which describe dynamics of viable cell density and concentrations of glucose, glutamine, ammonia, lactate, and the antibody fusion protein B1 (B1). The kinetic model has 18 parameters, six of which were calculated from the cell culture data, whereas the other 12 were estimated from a training data set that comprised of seven cell culture runs using a MCMC method. The model was confirmed in two validation data sets that represented a perturbation of the cell culture condition. The agreement between the predicted and measured values of both validation data sets may indicate high reliability of the model estimates. The kinetic model uniquely incorporated the ammonia removal and the exponential function of B1 protein concentration. The model indicated that ammonia and lactate play critical roles in cell growth and that low concentrations of glucose (0.17 mM) and glutamine (0.09 mM) in the cell culture medium may help reduce ammonia and lactate production. The model demonstrated that 83% of the glucose consumed was used for cell maintenance during the late phase of the cell cultures, whereas the maintenance coefficient for glutamine was negligible. Finally, the kinetic model suggests that it is critical for B1 production to sustain a high number of viable cells. The MCMC methodology may be a useful tool for modeling kinetics of a fed‐batch mammalian cell culture process. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2010     

Bayesian phylogenetic inference via Markov chain Monte Carlo methods

We derive a Markov chain to sample from the posterior distribution for a phylogenetic tree given sequence information from the corresponding set of organisms, a stochastic model for these data, and a prior distribution on the space of trees. A transformation of the tree into a canonical cophenetic matrix form suggests a simple and effective proposal distribution for selecting candidate trees close to the current tree in the chain. We illustrate the algorithm with restriction site data on 9 plant species, then extend to DNA sequences from 32 species of fish. The algorithm mixes well in both examples from random starting trees, generating reproducible estimates and credible sets for the path of evolution.     

Reversible jump Markov chain Monte Carlo computation and Bayesian model determination

Markov chain Monte Carlo methods for Bayesian computation haveuntil recently been restricted to problems where the joint distributionof all variables has a density with respect to some fixed standardunderlying measure. They have therefore not been available forapplication to Bayesian model determination, where the dimensionalityof the parameter vector is typically not fixed. This paper proposesa new framework for the construction of reversible Markov chainsamplers that jump between parameter subspaces of differingdimensionality, which is flexible and entirely constructive.It should therefore have wide applicability in model determinationproblems. The methodology is illustrated with applications tomultiple change-point analysis in one and two dimensions, andto a Bayesian comparison of binomial experiments.     

Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods

We introduce a new statistical computing method, called data cloning, to calculate maximum likelihood estimates and their standard errors for complex ecological models. Although the method uses the Bayesian framework and exploits the computational simplicity of the Markov chain Monte Carlo (MCMC) algorithms, it provides valid frequentist inferences such as the maximum likelihood estimates and their standard errors. The inferences are completely invariant to the choice of the prior distributions and therefore avoid the inherent subjectivity of the Bayesian approach. The data cloning method is easily implemented using standard MCMC software. Data cloning is particularly useful for analysing ecological situations in which hierarchical statistical models, such as state-space models and mixed effects models, are appropriate. We illustrate the method by fitting two nonlinear population dynamics models to data in the presence of process and observation noise.     

Comparison between estimation of breeding values and fixed effects using Bayesian and empirical BLUP estimation under selection on parents and missing pedigree information

Bayesian (via Gibbs sampling) and empirical BLUP (EBLUP) estimation of fixed effects and breeding values were compared by simulation. Combinations of two simulation models (with or without effect of contemporary group (CG)), three selection schemes (random, phenotypic and BLUP selection), two levels of heritability (0.20 and 0.50) and two levels of pedigree information (0% and 15% randomly missing) were considered. Populations consisted of 450 animals spread over six discrete generations. An infinitesimal additive genetic animal model was assumed while simulating data. EBLUP and Bayesian estimates of CG effects and breeding values were, in all situations, essentially the same with respect to Spearman''s rank correlation between true and estimated values. Bias and mean square error (MSE) of EBLUP and Bayesian estimates of CG effects and breeding values showed the same pattern over the range of simulated scenarios. Methods were not biased by phenotypic and BLUP selection when pedigree information was complete, albeit MSE of estimated breeding values increased for situations where CG effects were present. Estimation of breeding values by Bayesian and EBLUP was similarly affected by joint effect of phenotypic or BLUP selection and randomly missing pedigree information. For both methods, bias and MSE of estimated breeding values and CG effects substantially increased across generations.     

Markov chain Monte Carlo fitting of single-channel data from inositol trisphosphate receptors

In many cell types, the inositol trisphosphate receptor (IPR) is one of the important components that control intracellular calcium dynamics, and an understanding of this receptor (which is also a calcium channel) is necessary for an understanding of calcium oscillations and waves. Recent advances in experimental techniques now allow for the measurement of single-channel activity of the IPR in conditions similar to its native environment, and these data can be used to determine the rate constants in Markov models of the IPR. We illustrate a parameter estimation method based on Markov chain Monte Carlo, which can be used to fit directly to single-channel data, and determining, as an intrinsic part of the fit, the times at which the IPR is opening and closing. We show, using simulated data, the most complex Markov model that can be unambiguously determined from steady-state data and show that non-steady-state data is required to determine more complex models.     

A Markov chain Monte Carlo strategy for sampling from the joint posterior distribution of pedigrees and population parameters under a Fisher-Wright model with partial selfing

A simple population genetic model is presented for a hermaphrodite annual species, allowing both selfing and outcrossing. Those male gametes (pollen) responsible for outcrossing are assumed to disperse much further than seeds. Under this model, the pedigree of a sample from a single locality is loop-free. A novel Markov chain Monte Carlo strategy is presented for sampling from the joint posterior distribution of the pedigree of such a sample and the parameters of the population genetic model (including the selfing rate) given the genotypes of the sampled individuals at unlinked marker loci. The computational costs of this Markov chain Monte Carlo strategy scale well with the number of individuals in the sample, and the number of marker loci, but increase exponentially with the age (time since colonisation from the source population) of the local population. Consequently, this strategy is particularly suited to situations where the sample has been collected from a population which is the result of a recent colonisation process.     

Efficiency of Markov chain Monte Carlo tree proposals in Bayesian phylogenetics

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

A tutorial introduction to Bayesian inference for stochastic epidemic models using Markov chain Monte Carlo methods

Recent Bayesian methods for the analysis of infectious disease outbreak data using stochastic epidemic models are reviewed. These methods rely on Markov chain Monte Carlo methods. Both temporal and non-temporal data are considered. The methods are illustrated with a number of examples featuring different models and datasets.     

Influence of priors in Bayesian estimation of genetic parameters for multivariate threshold models using Gibbs sampling

Simulated data were used to investigate the influence of the choice of priors on estimation of genetic parameters in multivariate threshold models using Gibbs sampling. We simulated additive values, residuals and fixed effects for one continuous trait and liabilities of four binary traits, and QTL effects for one of the liabilities. Within each of four replicates six different datasets were generated which resembled different practical scenarios in horses with respect to number and distribution of animals with trait records and availability of QTL information. (Co)Variance components were estimated using a Bayesian threshold animal model via Gibbs sampling. The Gibbs sampler was implemented with both a flat and a proper prior for the genetic covariance matrix. Convergence problems were encountered in > 50% of flat prior analyses, with indications of potential or near posterior impropriety between about round 10 000 and 100 000. Terminations due to non-positive definite genetic covariance matrix occurred in flat prior analyses of the smallest datasets. Use of a proper prior resulted in improved mixing and convergence of the Gibbs chain. In order to avoid (near) impropriety of posteriors and extremely poorly mixing Gibbs chains, a proper prior should be used for the genetic covariance matrix when implementing the Gibbs sampler.     

Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models

Inference for Dirichlet process hierarchical models is typicallyperformed using Markov chain Monte Carlo methods, which canbe roughly categorized into marginal and conditional methods.The former integrate out analytically the infinite-dimensionalcomponent of the hierarchical model and sample from the marginaldistribution of the remaining variables using the Gibbs sampler.Conditional methods impute the Dirichlet process and updateit as a component of the Gibbs sampler. Since this requiresimputation of an infinite-dimensional process, implementationof the conditional method has relied on finite approximations.In this paper, we show how to avoid such approximations by designingtwo novel Markov chain Monte Carlo algorithms which sample fromthe exact posterior distribution of quantities of interest.The approximations are avoided by the new technique of retrospectivesampling. We also show how the algorithms can obtain samplesfrom functionals of the Dirichlet process. The marginal andthe conditional methods are compared and a careful simulationstudy is included, which involves a non-conjugate model, differentdatasets and prior specifications.     

Hierarchical Bayesian methods for estimation of parameters in a longitudinal HIV dynamic system

HIV dynamics studies have significantly contributed to the understanding of HIV infection and antiviral treatment strategies. But most studies are limited to short-term viral dynamics due to the difficulty of establishing a relationship of antiviral response with multiple treatment factors such as drug exposure and drug susceptibility during long-term treatment. In this article, a mechanism-based dynamic model is proposed for characterizing long-term viral dynamics with antiretroviral therapy, described by a set of nonlinear differential equations without closed-form solutions. In this model we directly incorporate drug concentration, adherence, and drug susceptibility into a function of treatment efficacy, defined as an inhibition rate of virus replication. We investigate a Bayesian approach under the framework of hierarchical Bayesian (mixed-effects) models for estimating unknown dynamic parameters. In particular, interest focuses on estimating individual dynamic parameters. The proposed methods not only help to alleviate the difficulty in parameter identifiability, but also flexibly deal with sparse and unbalanced longitudinal data from individual subjects. For illustration purposes, we present one simulation example to implement the proposed approach and apply the methodology to a data set from an AIDS clinical trial. The basic concept of the longitudinal HIV dynamic systems and the proposed methodologies are generally applicable to any other biomedical dynamic systems.