Model parameterization, prior distributions, and the general time-reversible model in Bayesian phylogenetics

Bayesian phylogenetic methods require the selection of prior probability distributions for all parameters of the model of evolution. These distributions allow one to incorporate prior information into a Bayesian analysis, but even in the absence of meaningful prior information, a prior distribution must be chosen. In such situations, researchers typically seek to choose a prior that will have little effect on the posterior estimates produced by an analysis, allowing the data to dominate. Sometimes a prior that is uniform (assigning equal prior probability density to all points within some range) is chosen for this purpose. In reality, the appropriate prior depends on the parameterization chosen for the model of evolution, a choice that is largely arbitrary. There is an extensive Bayesian literature on appropriate prior choice, and it has long been appreciated that there are parameterizations for which uniform priors can have a strong influence on posterior estimates. We here discuss the relationship between model parameterization and prior specification, using the general time-reversible model of nucleotide evolution as an example. We present Bayesian analyses of 10 simulated data sets obtained using a variety of prior distributions and parameterizations of the general time-reversible model. Uniform priors can produce biased parameter estimates under realistic conditions, and a variety of alternative priors avoid this bias.     

A hierarchical Bayesian model for calibrating estimates of species divergence times

In Bayesian divergence time estimation methods, incorporating calibrating information from the fossil record is commonly done by assigning prior densities to ancestral nodes in the tree. Calibration prior densities are typically parametric distributions offset by minimum age estimates provided by the fossil record. Specification of the parameters of calibration densities requires the user to quantify his or her prior knowledge of the age of the ancestral node relative to the age of its calibrating fossil. The values of these parameters can, potentially, result in biased estimates of node ages if they lead to overly informative prior distributions. Accordingly, determining parameter values that lead to adequate prior densities is not straightforward. In this study, I present a hierarchical Bayesian model for calibrating divergence time analyses with multiple fossil age constraints. This approach applies a Dirichlet process prior as a hyperprior on the parameters of calibration prior densities. Specifically, this model assumes that the rate parameters of exponential prior distributions on calibrated nodes are distributed according to a Dirichlet process, whereby the rate parameters are clustered into distinct parameter categories. Both simulated and biological data are analyzed to evaluate the performance of the Dirichlet process hyperprior. Compared with fixed exponential prior densities, the hierarchical Bayesian approach results in more accurate and precise estimates of internal node ages. When this hyperprior is applied using Markov chain Monte Carlo methods, the ages of calibrated nodes are sampled from mixtures of exponential distributions and uncertainty in the values of calibration density parameters is taken into account.     

Objective prior distributions for Jolly-Seber models of zero-augmented data

Statistical models of capture-recapture data that are used to estimate the dynamics of a population are known collectively as Jolly-Seber (JS) models. State-space versions of these models have been developed for the analysis of zero-augmented data that include the capture histories of the observed individuals and an arbitrarily large number of all-zero capture histories. The number of all-zero capture histories must be sufficiently large to include the unknown number N of individuals in the population that were ever alive during all sampling periods. This definition of N is equivalent to the “superpopulation” of individuals described in several JS models. To fit JS models of zero-augmented data, practitioners often assume a set of independent, uniform prior distributions for the recruitment parameters. However, if the number of capture histories is small compared to N, these uniform priors can exert considerable influence on the posterior distributions of N and other parameters because the uniform priors induce a highly skewed prior on N. In this article, I derive a class of prior distributions for the recruitment parameters of the JS model that can be used to specify objective prior distributions for N, including the discrete-uniform and the improper scale priors as special cases. This class of priors also may be used to specify prior knowledge about recruitment while still preserving the conditions needed to induce an objective prior on N. I use analyses of simulated and real data to illustrate the inferential benefits of this class of prior distributions and to identify circumstances where these benefits are most likely to be realized.     

Functional Uniform Priors for Nonlinear Modeling

Summary This article considers the topic of finding prior distributions when a major component of the statistical model depends on a nonlinear function. Using results on how to construct uniform distributions in general metric spaces, we propose a prior distribution that is uniform in the space of functional shapes of the underlying nonlinear function and then back‐transform to obtain a prior distribution for the original model parameters. The primary application considered in this article is nonlinear regression, but the idea might be of interest beyond this case. For nonlinear regression the so constructed priors have the advantage that they are parametrization invariant and do not violate the likelihood principle, as opposed to uniform distributions on the parameters or the Jeffrey’s prior, respectively. The utility of the proposed priors is demonstrated in the context of design and analysis of nonlinear regression modeling in clinical dose‐finding trials, through a real data example and simulation.     

Effectiveness of measures requiring and not requiring prior sequence alignment for estimating the dissimilarity of natural sequences

Summary Various measures of sequence dissimilarity have been evaluated by how well the additive least squares estimation of edges (branch lengths) of an unrooted evolutionary tree fit the observed pairwise dissimilarity measures and by how consistent the trees are for different data sets derived from the same set of sequences. This evaluation provided sensitive discrimination among dissimilarity measures and among possible trees. Dissimilarity measures not requiring prior sequence alignment did about as well as did the traditional mismatch counts requiring prior sequence alignment. Application of Jukes-Cantor correction to singlet mismatch counts worsened the results. Measures not requiring alignment had the advantage of being applicable to sequences too different to be critically alignable. Two different measures of pairwise dissimilarity not requiring alignment have been used: (1) multiplet distribution distance (MDD), the square of the Euclidean distance between vectors of the fractions of base singlets (or doublets, or triplets, or…) in the respective sequences, and (2) complements of long words (CLW), the count of bases not occurring in significantly long common words. MDD was applicable to sequences more different than was CLW (noncoding), but the latter often gave better results where both measures were available (coding). MDD results were improved by using longer multiplets and, if the sequences were coding, by using the larger amino acid and codon alphabets rather than the nucleotide alphabet. The additive least squares method could be used to provide a reasonable consensus of different trees for the same set of species (or related genes).     

Predicting Bald Eagle Collision at Wind Energy Facilities

Bald eagles (Haliaeetus leucocephalus) are currently protected in the United States under the Bald and Golden Eagle Protection Act of 1940 and Migratory Bird Treaty Act of 1918. Given these protections and the increasing development of wind energy throughout the United States, it is important for regulators and the wind industry to understand the risk of bald eagle collisions with wind turbines. Prior probability distributions for eagle exposure rates and collision rates have been developed for golden eagles (Aquila chrysaetos) by the United States Fish and Wildlife Service (USFWS). Given similar information has not been available for bald eagles, the current recommendation by the USFWS is to use the prior probability distributions developed using data collected on golden eagles to predict take for bald eagles. But some evidence suggests that bald and golden eagles may be at different risk for collision with wind turbines and the prior probability distributions developed for golden eagles may not be appropriate for bald eagles. We developed prior probability distributions using data collected at MidAmerican Energy Company's operating wind energy facilities in Iowa, USA, from December 2014 to March 2017 for bald eagle exposure rates and collision rates. The prior probability distribution for collision rate developed for bald eagles has a lower mean collision rate and less variability relative to that developed for golden eagles. We determined that the prior probability distributions specific to bald eagles from these operating facilities are a better starting point for predicting take for bald eagles at operating wind energy facilities in an agricultural landscape than those developed for golden eagles. © 2021 The Wildlife Society.     

Bayesian statistical analysis of protein side-chain rotamer preferences.

We present a Bayesian statistical analysis of the conformations of side chains in proteins from the Protein Data Bank. This is an extension of the backbone-dependent rotamer library, and includes rotamer populations and average chi angles for a full range of phi, psi values. The Bayesian analysis used here provides a rigorous statistical method for taking account of varying amounts of data. Bayesian statistics requires the assumption of a prior distribution for parameters over their range of possible values. This prior distribution can be derived from previous data or from pooling some of the present data. The prior distribution is combined with the data to form the posterior distribution, which is a compromise between the prior distribution and the data. For the chi 2, chi 3, and chi 4 rotamer prior distributions, we assume that the probability of each rotamer type is dependent only on the previous chi rotamer in the chain. For the backbone-dependence of the chi 1 rotamers, we derive prior distributions from the product of the phi-dependent and psi-dependent probabilities. Molecular mechanics calculations with the CHARMM22 potential show a strong similarity with the experimental distributions, indicating that proteins attain their lowest energy rotamers with respect to local backbone-side-chain interactions. The new library is suitable for use in homology modeling, protein folding simulations, and the refinement of X-ray and NMR structures.     

A Bayesian semiparametric accelerated failure time model

A Bayesian semiparametric approach is described for an accelerated failure time model. The error distribution is assigned a Pólya tree prior and the regression parameters a noninformative hierarchical prior. Two cases are considered: the first assumes error terms are exchangeable; the second assumes that error terms are partially exchangeable. A Markov chain Monte Carlo algorithm is described to obtain a predictive distribution for a future observation given both uncensored and censored data.     

Segregation ratios within Segregation Distorter lines of Drosophila melanogaster conform to a beta-binomial distribution

Segregation Distorter (SD) chromosomes are preferentially recovered from SD/SD+ males due to the dysfunction of sperm bearing the SD+ chromosome. The proportion of offspring bearing the SD chromosome is given the symbol k. The nature of the frequency distribution of k was examined by comparing observed k distributions produced by six different SD chromosomes, each with a different mean, with k distributions predicted by two different statistical models. The first model was one where the k of all males with a given SD chromosome were considered to be equal prior to the determination of those gametes which produce viable zygotes. In this model the only source of variation of k would be binomial sampling. The results rigorously demonstrated for the first time that the observed k distributions did not fit the prediction that the only source of variation was binomial sampling. The next model tested was that the prior distribution of segregation ratios conformed to a beta distribution, such that the distribution of k would be a beta-binomial distribution. The predicted distributions of this model did not differ significantly from the observed distributions of k in five of the six cases examined. The sixth case probably failed to fit a beta-binomial distribution due to a major segregating modifier. The demonstration that the prior distribution of segregation ratios of SD lines can generally be approximated with a beta distribution is crucial for the biometrical analysis of segregation distortion.     

Objective Bayesian analysis for the Student-t regression model

We develop a Bayesian analysis based on two different Jeffreyspriors for the Student-t regression model with unknown degreesof freedom. It is typically difficult to estimate the numberof degrees of freedom: improper prior distributions may leadto improper posterior distributions, whereas proper prior distributionsmay dominate the analysis. We show that Bayesian analysis witheither of the two considered Jeffreys priors provides a properposterior distribution. Finally, we show that Bayesian estimatorsbased on Jeffreys analysis compare favourably to other Bayesianestimators based on priors previously proposed in the literature.     

Using prior knowledge in the determination of macromolecular size-distributions by analytical ultracentrifugation

Analytical ultracentrifugation has reemerged as a widely used tool for the study of ensembles of biological macromolecules to understand, for example, their size-distribution and interactions in free solution. Such information can be obtained from the mathematical analysis of the concentration and signal gradients across the solution column and their evolution in time generated as a result of the gravitational force. In sedimentation velocity analytical ultracentrifugation, this analysis is frequently conducted using high resolution, diffusion-deconvoluted sedimentation coefficient distributions. They are based on Fredholm integral equations, which are ill-posed unless stabilized by regularization. In many fields, maximum entropy and Tikhonov-Phillips regularization are well-established and powerful approaches that calculate the most parsimonious distribution consistent with the data and prior knowledge, in accordance with Occam's razor. In the implementations available in analytical ultracentrifugation, to date, the basic assumption implied is that all sedimentation coefficients are equally likely and that the information retrieved should be condensed to the least amount possible. Frequently, however, more detailed distributions would be warranted by specific detailed prior knowledge on the macromolecular ensemble under study, such as the expectation of the sample to be monodisperse or paucidisperse or the expectation for the migration to establish a bimodal sedimentation pattern based on Gilbert-Jenkins' theory for the migration of chemically reacting systems. So far, such prior knowledge has remained largely unused in the calculation of the sedimentation coefficient or molecular weight distributions or was only applied as constraints. In the present paper, we examine how prior expectations can be built directly into the computational data analysis, conservatively in a way that honors the complete information of the experimental data, whether or not consistent with the prior expectation. Consistent with analogous results in other fields, we find that the use of available prior knowledge can have a dramatic effect on the resulting molecular weight, sedimentation coefficient, and size-and-shape distributions and can significantly increase both their sensitivity and their resolution. Further, the use of multiple alternative prior information allows us to probe the range of possible interpretations consistent with the data.     

Prior specification in Bayesian statistics: three cautionary tales

One of the most important differences between Bayesian and traditional techniques is that the former combines information available beforehand-captured in the prior distribution and reflecting the subjective state of belief before an experiment is carried out-and what the data teach us, as expressed in the likelihood function. Bayesian inference is based on the combination of prior and current information which is reflected in the posterior distribution. The fast growing implementation of Bayesian analysis techniques can be attributed to the development of fast computers and the availability of easy to use software. It has long been established that the specification of prior distributions should receive a lot of attention. Unfortunately, flat distributions are often (inappropriately) used in an automatic fashion in a wide range of types of models. We reiterate that the specification of the prior distribution should be done with great care and support this through three examples. Even in the absence of strong prior information, prior specification should be done at the appropriate scale of biological interest. This often requires incorporation of (weak) prior information based on common biological sense. Very weak and uninformative priors at one scale of the model may result in relatively strong priors at other levels affecting the posterior distribution. We present three different examples intu?vely illustrating this phenomenon indicating that this bias can be substantial (especially in small samples) and is widely present. We argue that complete ignorance or absence of prior information may not exist. Because the central theme of the Bayesian paradigm is to combine prior information with current data, authors should be encouraged to publish their raw data such that every scientist is able to perform an analysis incorporating his/her own (subjective) prior distributions.     

A whole-genome analysis using robust asymmetric distributions

This study is aimed at improving the analysis of data used in identifying marker-associated effects on quantitative traits, specifically to account for possible departures from a Gaussian distribution of the trait data and to allow for asymmetry of marker effects attributable to phenotypic divergence between parental lines. A Bayesian procedure for analysing marker effects at the whole-genome level is presented. The procedure adopts a skewed t-distribution as a prior distribution of marker effects. The model with the skewed t-process includes Gaussian prior distributions, skewed Gaussian prior distributions and symmetric t-distributions as special cases. A Markov Chain Monte Carlo algorithm for obtaining marginal posterior distributions of the unknowns is also presented. The method was applied to a dataset on three traits (live weight, carcass length and backfat depth) measured in an F2 cross between Iberian and Landrace pigs. The distribution of marker effects was clearly asymmetric for carcass length and backfat depth, whereas it was symmetric for live weight. The t-distribution seems more appropriate for describing the distribution of marker effects on backfat depth.     

Fragmentation gradients differentially affect the species range distributions of four taxonomic groups in semi‐deciduous Atlantic forest

Human activities often cause habitat fragmentation and how forest fragments affect species range distributions has implications for ecology and conservation. However, few studies have considered communities within the same landscape. Here, we analyzed metacommunity structure to determine the range distributions for species in four taxonomic groups (amphibians, birds, social wasps, and trees) in a patchy landscape of semi‐deciduous Atlantic forest in southwestern Brazil. Although trees are a key component of the environment for animals in forested patches, the ranges of bird, wasp, and amphibian species did not change in concert with the species ranges of trees. The species ranges of amphibians and social wasps were unaffected by fragmentation gradients and exhibited independent distribution patterns (i.e., random structure). In contrast, birds and trees exhibited range turnover along different fragmentation gradients, indicating that species show idiosyncratic responses to abiotic factors (i.e., Gleasonian structure). For birds, some less‐resilient species occurred only in fragments with a large area of native vegetation at a radius of 5 km from the center of the sampled forest fragments, whereas other more stress‐tolerant species occurred only in sites with small areas of native vegetation. For trees, some later succession species (e.g., animal‐dispersed seeds) occurred only in fragments with high connectivity, whereas earlier‐recruiting species (e.g., wind‐dispersed seeds) occurred in fragments with low connectivity. Thus, determining the effects of human‐modified landscapes on species range distributions, even within the same landscape, might not be a trivial task.     

Potential Changes in the Distributions of Western North America Tree and Shrub Taxa under Future Climate Scenarios

Increases in atmospheric greenhouse gases are driving significant changes in global climate. To project potential vegetation response to future climate change, this study uses response surfaces to describe the relationship between bioclimatic variables and the distribution of tree and shrub taxa in western North America. The response surfaces illustrate the probability of the occurrence of a taxon at particular points in climate space. Climate space was defined using three bioclimatic variables: mean temperature of the coldest month, growing degree days, and a moisture index. Species distributions were simulated under present climate using observed data (1951–80, 30-year mean) and under future climate (2090–99, 10-year mean) using scenarios generated by three general circulation models—HADCM2, CGCM1, and CSIRO. The scenarios assume a 1% per year compound increase in greenhouse gases and changes in sulfate (SO₄) aerosols based on the Intergovernmental Panel on Climate Change (IPCC) IS92a scenario. The results indicate that under future climate conditions, potential range changes could be large for many tree and shrub taxa. Shifts in the potential ranges of species are simulated to occur not only northward but in all directions, including southward of the existing ranges of certain species. The simulated potential distributions of some species become increasingly fragmented under the future climate scenarios, while the simulated potential distributions of other species expand. The magnitudes of the simulated range changes imply significant impacts to ecosystems and shifts in patterns of species diversity in western North America. Received 12 May 2000; accepted 20 December 2000.     

Separating Components of Variation in Survival of Mule Deer in Colorado

ABSTRACT Survival is an important parameter for understanding population dynamics of mule deer (Odocoileus hemionus) and other large herbivores. To understand long-term dynamics it is important to separate sampling and biological process variation in survival. Moreover, knowledge of correlations in survival across space and between young and adults can provide more informed predictions of survival in unsampled areas. We estimated survival of fawn, yearling, and adult mule deer from 4 spatially separated regions of Colorado, USA, from 1997 to 2008. We also estimated process variance in survival across time for each age and site using Markov chain Monte Carlo (MCMC) methods. Finally, we estimated correlations in survival among sites and ages with MCMC methods. Average winter fawn survival was 0.721 (SD = 0.024) for the 4 regions. Average winter adult female survival was 0.935 (SD = 0.007). Annual adult female survival ranged from 0.803 (SD = 0.017) to 0.900 (SD = 0.028) for the 4 regions, excluding hunting mortality. The correlation between fawn and adult female survival was high, 0.563 (SD = 0.253). Correlations in winter fawn survival were higher between populations at the same latitude than they were for populations to the north and south. We used survival estimates from our analysis to inform prior distributions for a Bayesian population dynamics model from one population in Colorado and compared that model to one with noninformative prior distributions. Population models including informative prior distributions based on our results performed better than those noninformative prior distributions on survival, providing more biologically defensible results when data were sparse. Knowledge of process distributions of survival can help wildlife managers better predict future population status and understand the likely range of survival rates.     

Using the Pareto principle in genome-wide breeding value estimation

Genome-wide breeding value (GWEBV) estimation methods can be classified based on the prior distribution assumptions of marker effects. Genome-wide BLUP methods assume a normal prior distribution for all markers with a constant variance, and are computationally fast. In Bayesian methods, more flexible prior distributions of SNP effects are applied that allow for very large SNP effects although most are small or even zero, but these prior distributions are often also computationally demanding as they rely on Monte Carlo Markov chain sampling. In this study, we adopted the Pareto principle to weight available marker loci, i.e., we consider that x% of the loci explain (100 - x)% of the total genetic variance. Assuming this principle, it is also possible to define the variances of the prior distribution of the ''big'' and ''small'' SNP. The relatively few large SNP explain a large proportion of the genetic variance and the majority of the SNP show small effects and explain a minor proportion of the genetic variance. We name this method MixP, where the prior distribution is a mixture of two normal distributions, i.e. one with a big variance and one with a small variance. Simulation results, using a real Norwegian Red cattle pedigree, show that MixP is at least as accurate as the other methods in all studied cases. This method also reduces the hyper-parameters of the prior distribution from 2 (proportion and variance of SNP with big effects) to 1 (proportion of SNP with big effects), assuming the overall genetic variance is known. The mixture of normal distribution prior made it possible to solve the equations iteratively, which greatly reduced computation loads by two orders of magnitude. In the era of marker density reaching million(s) and whole-genome sequence data, MixP provides a computationally feasible Bayesian method of analysis.     

Are animals capable of Bayesian updating? An empirical review

Numerous behavioral models assume individuals combine knowledge in the form of a prior distribution with current sample information using Bayesian updating to estimate the quality of environmental parameters. I examine this assumption by reviewing 11 empirical studies. Six studies compared observed behavior to predictions of Bayesian and non-Bayesian models, while five studies manipulated prior distributions directly and observed how such manipulations altered behavior. Eight species of birds, three mammals, one fish and one insect exhibited behavior consistent with Bayesian updating models; one studied bird species failed to show evidence of Bayesian updating. Most studies examined how individuals estimated food patch quality but two investigated mating decisions. These studies suggest a variety of animals in different ecological contexts behave in manners consistent with predictions of Bayesian updating models. Future work on decision-making should focus on understanding how animals learn prior distributions and on decision-making in additional ecological contexts.     

Cytokinetic properties of asynchronous and cytosine arabinoside perturbed murine tumors measured by simultaneous bromodeoxyuridine/DNA analyses

This paper describes the determination of cytokinetic properties of asynchronous and cytosine arabinoside- (Ara-C) treated KHT tumors growing in vivo using the bromodeoxyuridine (BrdUrd)/DNA analysis technique. The cytokinetic properties of asynchronously growing tumors were estimated by computer analysis of sequential BrdUrd/DNA distributions measured at 2- to 3-h intervals after administration of a single i.p. injection of BrdUrd. The cytokinetic properties of the Ara-C-treated tumors were estimated by computer analysis of BrdUrd/DNA distributions measured at 2- to 3-h intervals after Ara-C treatment. BrdUrd was injected 30 min prior to tumor harvest. The cytokinetic properties of the cells rendered nonclonogenic by Ara-C were followed in BrdUrd/DNA distributions measured at 2- to 3-h intervals after Ara-C treatment of tumors that were labeled with BrdUrd 30 min prior to Ara C injection. The G1-, S-, and G2M-phase durations were estimated to be 7.6, 10.9, and 2.0 h prior to Ara-C; decreasing to 1.2, 4.1, and 1.4 after Ara-C. The growth fraction was estimated to be 0.8 prior to Ara-C. Complete recruitment of the normally noncycling subpopulation was observed after Ara-C treatment. Ara-C-killed cells were removed from the tumor within 24 h following Ara-C injection. These cytokinetic properties were similar to those reported in other studies.