Parallel Metropolis coupled Markov chain Monte Carlo for Bayesian phylogenetic inference

MOTIVATION: Bayesian estimation of phylogeny is based on the posterior probability distribution of trees. Currently, the only numerical method that can effectively approximate posterior probabilities of trees is Markov chain Monte Carlo (MCMC). Standard implementations of MCMC can be prone to entrapment in local optima. Metropolis coupled MCMC [(MC)(3)], a variant of MCMC, allows multiple peaks in the landscape of trees to be more readily explored, but at the cost of increased execution time. RESULTS: This paper presents a parallel algorithm for (MC)(3). The proposed parallel algorithm retains the ability to explore multiple peaks in the posterior distribution of trees while maintaining a fast execution time. The algorithm has been implemented using two popular parallel programming models: message passing and shared memory. Performance results indicate nearly linear speed improvement in both programming models for small and large data sets.     

Estimating species trees using multiple-allele DNA sequence data

Several techniques, such as concatenation and consensus methods, are available for combining data from multiple loci to produce a single statement of phylogenetic relationships. However, when multiple alleles are sampled from individual species, it becomes more challenging to estimate relationships at the level of species, either because concatenation becomes inappropriate due to conflicts among individual gene trees, or because the species from which multiple alleles have been sampled may not form monophyletic groups in the estimated tree. We propose a Bayesian hierarchical model to reconstruct species trees from multiple-allele, multilocus sequence data, building on a recently proposed method for estimating species trees from single allele multilocus data. A two-step Markov Chain Monte Carlo (MCMC) algorithm is adopted to estimate the posterior distribution of the species tree. The model is applied to estimate the posterior distribution of species trees for two multiple-allele datasets--yeast (Saccharomyces) and birds (Manacus-manakins). The estimates of the species trees using our method are consistent with those inferred from other methods and genetic markers, but in contrast to other species tree methods, it provides credible regions for the species tree. The Bayesian approach described here provides a powerful framework for statistical testing and integration of population genetics and phylogenetics.     

Gaussian Process Based Bayesian Semiparametric Quantitative Trait Loci Interval Mapping

Summary .  In linkage analysis, it is often necessary to include covariates such as age or weight to increase power or avoid spurious false positive findings. However, if a covariate term in the model is specified incorrectly (e.g., a quadratic term misspecified as a linear term), then the inclusion of the covariate may adversely affect power and accuracy of the identification of quantitative trait loci (QTL). Furthermore, some covariates may interact with each other in a complicated fashion. We implement semiparametric models for single and multiple QTL mapping. Both mapping methods include an unspecified function of any covariate found or suspected to have a more complex than linear but unknown relationship with the response variable. They also allow for interactions among different covariates. This analysis is performed in a Bayesian inference framework using Markov chain Monte Carlo. The advantages of our methods are demonstrated via extensive simulations and real data analysis.     

A unified Markov chain Monte Carlo framework for mapping multiple quantitative trait loci

In this article, a unified Markov chain Monte Carlo (MCMC) framework is proposed to identify multiple quantitative trait loci (QTL) for complex traits in experimental designs, based on a composite space representation of the problem that has fixed dimension. The proposed unified approach includes the existing Bayesian QTL mapping methods using reversible jump MCMC algorithm as special cases. We also show that a variety of Bayesian variable selection methods using Gibbs sampling can be applied to the composite model space for mapping multiple QTL. The unified framework not only results in some new algorithms, but also gives useful insight into some of the important factors governing the performance of Gibbs sampling and reversible jump for mapping multiple QTL. Finally, we develop strategies to improve the performance of MCMC algorithms.     

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 hierarchical semiparametric regression model for combining HIV-1 phylogenetic analyses using iterative reweighting algorithms

Phylogenetic modeling is computationally challenging and most phylogeny models fit a single phylogeny to a single set of molecular sequences. Individual phylogenetic analyses are typically performed independently using publicly available software that fits a computationally intensive Bayesian model using Markov chain Monte Carlo (MCMC) simulation. We develop a Bayesian hierarchical semiparametric regression model to combine multiple phylogenetic analyses of HIV-1 nucleotide sequences and estimate parameters of interest within and across analyses. We use a mixture of Dirichlet processes as a prior for the parameters to relax inappropriate parametric assumptions and to ensure the prior distribution for the parameters is continuous. We use several reweighting algorithms for combining completed MCMC analyses to shrink parameter estimates while adjusting for data set-specific covariates. This avoids constructing a large complex model involving all the original data, which would be computationally challenging and would require rewriting the existing stand-alone software.     

Frequentist properties of Bayesian posterior probabilities of phylogenetic trees under simple and complex substitution models

What does the posterior probability of a phylogenetic tree mean?This simulation study shows that Bayesian posterior probabilities have the meaning that is typically ascribed to them; the posterior probability of a tree is the probability that the tree is correct, assuming that the model is correct. At the same time, the Bayesian method can be sensitive to model misspecification, and the sensitivity of the Bayesian method appears to be greater than the sensitivity of the nonparametric bootstrap method (using maximum likelihood to estimate trees). Although the estimates of phylogeny obtained by use of the method of maximum likelihood or the Bayesian method are likely to be similar, the assessment of the uncertainty of inferred trees via either bootstrapping (for maximum likelihood estimates) or posterior probabilities (for Bayesian estimates) is not likely to be the same. We suggest that the Bayesian method be implemented with the most complex models of those currently available, as this should reduce the chance that the method will concentrate too much probability on too few trees.     

Multipoint mapping of viability and segregation distorting loci using molecular markers

In line-crossing experiments, deviations from Mendelian segregation ratios are usually observed for some markers. We hypothesize that these deviations are caused by one or more segregation-distorting loci (SDL) linked to the markers. We develop both a maximum-likelihood (ML) method and a Bayesian method to map SDL using molecular markers. The ML mapping is implemented via an EM algorithm and the Bayesian method is performed via the Markov chain Monte Carlo (MCMC). The Bayesian mapping is computationally more intensive than the ML mapping but can handle more complicated models such as multiple SDL and variable number of SDL. Both methods are applied to a set of simulated data and real data from a cross of two Scots pine trees.     

Gene tree parsimony of multilocus snake venom protein families reveals species tree conflict as a result of multiple parallel gene loss

The proliferation of gene data from multiple loci of large multigene families has been greatly facilitated by considerable recent advances in sequence generation. The evolution of such gene families, which often undergo complex histories and different rates of change, combined with increases in sequence data, pose complex problems for traditional phylogenetic analyses, and in particular, those that aim to successfully recover species relationships from gene trees. Here, we implement gene tree parsimony analyses on multicopy gene family data sets of snake venom proteins for two separate groups of taxa, incorporating Bayesian posterior distributions as a rigorous strategy to account for the uncertainty present in gene trees. Gene tree parsimony largely failed to infer species trees congruent with each other or with species phylogenies derived from mitochondrial and single-copy nuclear sequences. Analysis of four toxin gene families from a large expressed sequence tag data set from the viper genus Echis failed to produce a consistent topology, and reanalysis of a previously published gene tree parsimony data set, from the family Elapidae, suggested that species tree topologies were predominantly unsupported. We suggest that gene tree parsimony failure in the family Elapidae is likely the result of unequal and/or incomplete sampling of paralogous genes and demonstrate that multiple parallel gene losses are likely responsible for the significant species tree conflict observed in the genus Echis. These results highlight the potential for gene tree parsimony analyses to be undermined by rapidly evolving multilocus gene families under strong natural selection.     

Bayesian analysis of binary prediction tree models for retrospectively sampled outcomes

Classification tree models are flexible analysis tools which have the ability to evaluate interactions among predictors as well as generate predictions for responses of interest. We describe Bayesian analysis of a specific class of tree models in which binary response data arise from a retrospective case-control design. We are also particularly interested in problems with potentially very many candidate predictors. This scenario is common in studies concerning gene expression data, which is a key motivating example context. Innovations here include the introduction of tree models that explicitly address and incorporate the retrospective design, and the use of nonparametric Bayesian models involving Dirichlet process priors on the distributions of predictor variables. The model specification influences the generation of trees through Bayes' factor based tests of association that determine significant binary partitions of nodes during a process of forward generation of trees. We describe this constructive process and discuss questions of generating and combining multiple trees via Bayesian model averaging for prediction. Additional discussion of parameter selection and sensitivity is given in the context of an example which concerns prediction of breast tumour status utilizing high-dimensional gene expression data; the example demonstrates the exploratory/explanatory uses of such models as well as their primary utility in prediction. Shortcomings of the approach and comparison with alternative tree modelling algorithms are also discussed, as are issues of modelling and computational extensions.     

Cumulative disease progression models for cross‐sectional data: A review and comparison

A better understanding of disease progression is beneficial for early diagnosis and appropriate individual therapy. Many different approaches for statistical modelling of cumulative disease progression have been proposed in the literature, including simple path models up to complex restricted Bayesian networks. Important fields of application are diseases such as cancer and HIV. Tumour progression is measured by means of chromosome aberrations, whereas people infected with HIV develop drug resistances because of genetic changes of the HI‐virus. These two very different diseases have typical courses of disease progression, which can be modelled partly by consecutive and partly by independent steps. This paper gives an overview of the different progression models and points out their advantages and drawbacks. Different models are compared via simulations to analyse how they work if some of their assumptions are violated. In a simulation study, we evaluate how models perform in terms of fitting induced multivariate probability distributions and topological relationships. We often find that the true model class used for generating data is outperformed by either a less or a more complex model class. The more flexible conjunctive Bayesian networks can be used to fit oncogenetic trees, whereas mixtures of oncogenetic trees with three tree components can be well fitted by mixture models with only two tree components.     

Linearized Forms of Individual-Level Models for Large-Scale Spatial Infectious Disease Systems

Individual-level models (ILMs) for infectious diseases, fitted in a Bayesian MCMC framework, are an intuitive and flexible class of models that can take into account population heterogeneity via various individual-level covariates. ILMs containing a geometric distance kernel to account for geographic heterogeneity provide a natural way to model the spatial spread of many diseases. However, in even only moderately large populations, the likelihood calculations required can be prohibitively time consuming. It is possible to speed up the computation via a technique which makes use a linearized distance kernel. Here we examine some methods of carrying out this linearization and compare the performances of these methods.     

Bayesian Mapping of Lichens Growing on Trees

Suitability of trees as hosts for epiphytic lichens are studied in a forest stand of size 25 ha. Suitability is measured as occupation probabilites which are modelled using hierarchical Bayesian approach. These probabilities are useful for an ecologist. They give smoothed spatial distribution map of suitability for each of the species and can be used in detecting high‐ and low‐probability areas. In addition, suitability is explained by tree‐level covariates. Spatial dependence, which is due to unobserved spatially structured covariates, is modelled through an unobserved Markov random field. Markov chain Monte Carlo method has been applied in Bayesian computation. The extensive spatial data consist of the occurrences of eight lichen species and one bryophyte on all of the 1253 potential host trees. In addition, coordinates of the trees and several tree characteristics have been recorded. The data have been analysed for four most abundant species: Lobaria pulmonaria, Nephroma bellum, Nephroma parile and Peltigera praetextata. The tree level parameters, subject to estimation, consist of the occurrence probabilities for each tree and for each lichen species. Model validation is discussed in detail and, in addition to Bayesian validation tools, the autologistic model and case‐control design based on logistic regression have been suggested for validation of covariate effects. As a result we present suitability maps for the four lichen species. We observed, that among the observed tree covariates, the diameter at breast height (DBH) correlates with lichen occurrence. Our modelling approach has close connections to disease mapping in spatial epidemiology.     

Time-varying Markov regression random-effect model with Bayesian estimation procedures: Application to dynamics of functional recovery in patients with stroke

The rates of functional recovery after stroke tend to decrease with time. Time-varying Markov processes (TVMP) may be more biologically plausible than time-invariant Markov process for modeling such data. However, analysis of such stochastic processes, particularly tackling reversible transitions and the incorporation of random effects into models, can be analytically intractable. We make use of ordinary differential equations to solve continuous-time TVMP with reversible transitions. The proportional hazard form was used to assess the effects of an individual’s covariates on multi-state transitions with the incorporation of random effects that capture the residual variation after being explained by measured covariates under the concept of generalized linear model. We further built up Bayesian directed acyclic graphic model to obtain full joint posterior distribution. Markov chain Monte Carlo (MCMC) with Gibbs sampling was applied to estimate parameters based on posterior marginal distributions with multiple integrands. The proposed method was illustrated with empirical data from a study on the functional recovery after stroke.     

Guided tree topology proposals for Bayesian phylogenetic inference

Increasingly, large data sets pose a challenge for computationally intensive phylogenetic methods such as Bayesian Markov chain Monte Carlo (MCMC). Here, we investigate the performance of common MCMC proposal distributions in terms of median and variance of run time to convergence on 11 data sets. We introduce two new Metropolized Gibbs Samplers for moving through "tree space." MCMC simulation using these new proposals shows faster average run time and dramatically improved predictability in performance, with a 20-fold reduction in the variance of the time to estimate the posterior distribution to a given accuracy. We also introduce conditional clade probabilities and demonstrate that they provide a superior means of approximating tree topology posterior probabilities from samples recorded during MCMC.     

Species interactions weakly modify climate‐induced tree co‐occurrence patterns

Aims

Species distributions are hypothesized to be underlain by a complex association of processes that span multiple spatial scales including biotic interactions, dispersal limitation, fine‐scale resource gradients and climate. Species disequilibrium with climate may reflect the effects of non‐climatic processes on species distributions, yet distribution models have rarely directly considered non‐climatic processes. Here, we use a Joint Species Distribution Model (JSDM) to investigate the influence of non‐climatic factors on species co‐occurrence patterns and to directly quantify the relative influences of climate and alternative processes that may generate correlated responses in species distributions, such as species interactions, on tree co‐occurrence patterns.

Location

US Rocky Mountains.

Methods

We apply a Bayesian JSDM to simultaneously model the co‐occurrence patterns of ten dominant tree species across the Rocky Mountains, and evaluate climatic and residual correlations from the fitted model to determine the relative contribution of each component to observed co‐occurrence patterns. We also evaluate predictions generated from the fitted model relative to a single‐species modelling approach.

Results

For most species, correlation due to climate covariates exceeded residual correlation, indicating an overriding influence of broad‐scale climate on co‐occurrence patterns. Accounting for covariance among species did not significantly improve predictions relative to a single‐species approach, providing limited evidence for a strong independent influence of species interactions on distribution patterns.

Conclusions

Overall, our findings indicate that climate is an important driver of regional biodiversity patterns and that interactions between dominant tree species contribute little to explain species co‐occurrence patterns among Rocky Mountain trees.     

Accurate Phylogenetic Tree Reconstruction from Quartets: A Heuristic Approach

Supertree methods construct trees on a set of taxa (species) combining many smaller trees on the overlapping subsets of the entire set of taxa. A ‘quartet’ is an unrooted tree over taxa, hence the quartet-based supertree methods combine many -taxon unrooted trees into a single and coherent tree over the complete set of taxa. Quartet-based phylogeny reconstruction methods have been receiving considerable attentions in the recent years. An accurate and efficient quartet-based method might be competitive with the current best phylogenetic tree reconstruction methods (such as maximum likelihood or Bayesian MCMC analyses), without being as computationally intensive. In this paper, we present a novel and highly accurate quartet-based phylogenetic tree reconstruction method. We performed an extensive experimental study to evaluate the accuracy and scalability of our approach on both simulated and biological datasets.     

Bayesian mapping of quantitative trait loci for multiple complex traits with the use of variance components

Complex traits important for humans are often correlated phenotypically and genetically. Joint mapping of quantitative-trait loci (QTLs) for multiple correlated traits plays an important role in unraveling the genetic architecture of complex traits. Compared with single-trait analysis, joint mapping addresses more questions and has advantages for power of QTL detection and precision of parameter estimation. Some statistical methods have been developed to map QTLs underlying multiple traits, most of which are based on maximum-likelihood methods. We develop here a multivariate version of the Bayes methodology for joint mapping of QTLs, using the Markov chain-Monte Carlo (MCMC) algorithm. We adopt a variance-components method to model complex traits in outbred populations (e.g., humans). The method is robust, can deal with an arbitrary number of alleles with arbitrary patterns of gene actions (such as additive and dominant), and allows for multiple phenotype data of various types in the joint analysis (e.g., multiple continuous traits and mixtures of continuous traits and discrete traits). Under a Bayesian framework, parameters--including the number of QTLs--are estimated on the basis of their marginal posterior samples, which are generated through two samplers, the Gibbs sampler and the reversible-jump MCMC. In addition, we calculate the Bayes factor related to each identified QTL, to test coincident linkage versus pleiotropy. The performance of our method is evaluated in simulations with full-sib families. The results show that our proposed Bayesian joint-mapping method performs well for mapping multiple QTLs in situations of either bivariate continuous traits or mixed data types. Compared with the analysis for each trait separately, Bayesian joint mapping improves statistical power, provides stronger evidence of QTL detection, and increases precision in estimation of parameter and QTL position. We also applied the proposed method to a set of real data and detected a coincident linkage responsible for determining bone mineral density and areal bone size of wrist in humans.     

Bayesian Mixture Models for Assessment of Gene Differential Behaviour and Prediction of pCR through the Integration of Copy Number and Gene Expression Data

We consider modeling jointly microarray RNA expression and DNA copy number data. We propose Bayesian mixture models that define latent Gaussian probit scores for the DNA and RNA, and integrate between the two platforms via a regression of the RNA probit scores on the DNA probit scores. Such a regression conveniently allows us to include additional sample specific covariates such as biological conditions and clinical outcomes. The two developed methods are aimed respectively to make inference on differential behaviour of genes in patients showing different subtypes of breast cancer and to predict the pathological complete response (pCR) of patients borrowing strength across the genomic platforms. Posterior inference is carried out via MCMC simulations. We demonstrate the proposed methodology using a published data set consisting of 121 breast cancer patients.     

Bayesian phylogenetic analysis of combined data

The recent development of Bayesian phylogenetic inference using Markov chain Monte Carlo (MCMC) techniques has facilitated the exploration of parameter-rich evolutionary models. At the same time, stochastic models have become more realistic (and complex) and have been extended to new types of data, such as morphology. Based on this foundation, we developed a Bayesian MCMC approach to the analysis of combined data sets and explored its utility in inferring relationships among gall wasps based on data from morphology and four genes (nuclear and mitochondrial, ribosomal and protein coding). Examined models range in complexity from those recognizing only a morphological and a molecular partition to those having complex substitution models with independent parameters for each gene. Bayesian MCMC analysis deals efficiently with complex models: convergence occurs faster and more predictably for complex models, mixing is adequate for all parameters even under very complex models, and the parameter update cycle is virtually unaffected by model partitioning across sites. Morphology contributed only 5% of the characters in the data set but nevertheless influenced the combined-data tree, supporting the utility of morphological data in multigene analyses. We used Bayesian criteria (Bayes factors) to show that process heterogeneity across data partitions is a significant model component, although not as important as among-site rate variation. More complex evolutionary models are associated with more topological uncertainty and less conflict between morphology and molecules. Bayes factors sometimes favor simpler models over considerably more parameter-rich models, but the best model overall is also the most complex and Bayes factors do not support exclusion of apparently weak parameters from this model. Thus, Bayes factors appear to be useful for selecting among complex models, but it is still unclear whether their use strikes a reasonable balance between model complexity and error in parameter estimates.