Bayesian Estimation of the Number of Species using Noninformative Priors

Consider a sample of animal abundances collected from one sampling occasion. Our focus is in estimating the number of species in a closed population. In order to conduct a noninformative Bayesian inference when modeling this data, we derive Jeffreys and reference priors from the full likelihood. We assume that the species' abundances are randomly distributed according to a distribution indexed by a finite‐dimensional parameter. We consider two specific cases which assume that the mean abundances are constant or exponentially distributed. The Jeffreys and reference priors are functions of the Fisher information for the model parameters; the information is calculated in part using the linear difference score for integer parameter models (Lindsay & Roeder 1987). The Jeffreys and reference priors perform similarly in a data example we consider. The posteriors based on the Jeffreys and reference priors are proper. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A cautionary note on Bayesian estimation of population size by removal sampling with diffuse priors

We consider the problem of estimating a population size by removal sampling when the sampling rate is unknown. Bayesian methods are now widespread and allow to include prior knowledge in the analysis. However, we show that Bayes estimates based on default improper priors lead to improper posteriors or infinite estimates. Similarly, weakly informative priors give unstable estimators that are sensitive to the choice of hyperparameters. By examining the likelihood, we show that population size estimates can be stabilized by penalizing small values of the sampling rate or large value of the population size. Based on theoretical results and simulation studies, we propose some recommendations on the choice of the prior. Then, we applied our results to real datasets.     

Frequentist performance of Bayesian confidence intervals for comparing proportions in 2 x 2 contingency tables

This article investigates the performance, in a frequentist sense, of Bayesian confidence intervals (CIs) for the difference of proportions, relative risk, and odds ratio in 2 x 2 contingency tables. We consider beta priors, logit-normal priors, and related correlated priors for the two binomial parameters. The goal was to analyze whether certain settings for prior parameters tend to provide good coverage performance regardless of the true association parameter values. For the relative risk and odds ratio, we recommend tail intervals over highest posterior density (HPD) intervals, for invariance reasons. To protect against potentially very poor coverage probabilities when the effect is large, it is best to use a diffuse prior, and we recommend the Jeffreys prior. Otherwise, with relatively small samples, Bayesian CIs using more informative (even uniform) priors tend to have poorer performance than the frequentist CIs based on inverting score tests, which perform uniformly quite well for these parameters.     

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.     

Nonconjugate Bayesian analysis of variance component models

We consider the usual normal linear mixed model for variance components from a Bayesian viewpoint. With conjugate priors and balanced data, Gibbs sampling is easy to implement; however, simulating from full conditionals can become difficult for the analysis of unbalanced data with possibly nonconjugate priors, thus leading one to consider alternative Markov chain Monte Carlo schemes. We propose and investigate a method for posterior simulation based on an independence chain. The method is customized to exploit the structure of the variance component model, and it works with arbitrary prior distributions. As a default reference prior, we use a version of Jeffreys' prior based on the integrated (restricted) likelihood. We demonstrate the ease of application and flexibility of this approach in familiar settings involving both balanced and unbalanced 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.     

Tree Biomass Estimation of Chinese fir (Cunninghamia lanceolata) Based on Bayesian Method

Chinese fir (Cunninghamia lanceolata (Lamb.) Hook.) is the most important conifer species for timber production with huge distribution area in southern China. Accurate estimation of biomass is required for accounting and monitoring Chinese forest carbon stocking. In the study, allometric equation was used to analyze tree biomass of Chinese fir. The common methods for estimating allometric model have taken the classical approach based on the frequency interpretation of probability. However, many different biotic and abiotic factors introduce variability in Chinese fir biomass model, suggesting that parameters of biomass model are better represented by probability distributions rather than fixed values as classical method. To deal with the problem, Bayesian method was used for estimating Chinese fir biomass model. In the Bayesian framework, two priors were introduced: non-informative priors and informative priors. For informative priors, 32 biomass equations of Chinese fir were collected from published literature in the paper. The parameter distributions from published literature were regarded as prior distributions in Bayesian model for estimating Chinese fir biomass. Therefore, the Bayesian method with informative priors was better than non-informative priors and classical method, which provides a reasonable method for estimating Chinese fir biomass.     

On the Origins of Suboptimality in Human Probabilistic Inference

Humans have been shown to combine noisy sensory information with previous experience (priors), in qualitative and sometimes quantitative agreement with the statistically-optimal predictions of Bayesian integration. However, when the prior distribution becomes more complex than a simple Gaussian, such as skewed or bimodal, training takes much longer and performance appears suboptimal. It is unclear whether such suboptimality arises from an imprecise internal representation of the complex prior, or from additional constraints in performing probabilistic computations on complex distributions, even when accurately represented. Here we probe the sources of suboptimality in probabilistic inference using a novel estimation task in which subjects are exposed to an explicitly provided distribution, thereby removing the need to remember the prior. Subjects had to estimate the location of a target given a noisy cue and a visual representation of the prior probability density over locations, which changed on each trial. Different classes of priors were examined (Gaussian, unimodal, bimodal). Subjects'' performance was in qualitative agreement with the predictions of Bayesian Decision Theory although generally suboptimal. The degree of suboptimality was modulated by statistical features of the priors but was largely independent of the class of the prior and level of noise in the cue, suggesting that suboptimality in dealing with complex statistical features, such as bimodality, may be due to a problem of acquiring the priors rather than computing with them. We performed a factorial model comparison across a large set of Bayesian observer models to identify additional sources of noise and suboptimality. Our analysis rejects several models of stochastic behavior, including probability matching and sample-averaging strategies. Instead we show that subjects'' response variability was mainly driven by a combination of a noisy estimation of the parameters of the priors, and by variability in the decision process, which we represent as a noisy or stochastic posterior.     

Moving beyond noninformative priors: why and how to choose weakly informative priors in Bayesian analyses

Throughout the last two decades, Bayesian statistical methods have proliferated throughout ecology and evolution. Numerous previous references established both philosophical and computational guidelines for implementing Bayesian methods. However, protocols for incorporating prior information, the defining characteristic of Bayesian philosophy, are nearly nonexistent in the ecological literature. Here, I hope to encourage the use of weakly informative priors in ecology and evolution by providing a ‘consumer's guide’ to weakly informative priors. The first section outlines three reasons why ecologists should abandon noninformative priors: 1) common flat priors are not always noninformative, 2) noninformative priors provide the same result as simpler frequentist methods, and 3) noninformative priors suffer from the same high type I and type M error rates as frequentist methods. The second section provides a guide for implementing informative priors, wherein I detail convenient ‘reference’ prior distributions for common statistical models (i.e. regression, ANOVA, hierarchical models). I then use simulations to visually demonstrate how informative priors influence posterior parameter estimates. With the guidelines provided here, I hope to encourage the use of weakly informative priors for Bayesian analyses in ecology. Ecologists can and should debate the appropriate form of prior information, but should consider weakly informative priors as the new ‘default’ prior for any Bayesian model.     

Tail paradox, partial identifiability, and influential priors in Bayesian branch length inference

Recent studies have observed that Bayesian analyses of sequence data sets using the program MrBayes sometimes generate extremely large branch lengths, with posterior credibility intervals for the tree length (sum of branch lengths) excluding the maximum likelihood estimates. Suggested explanations for this phenomenon include the existence of multiple local peaks in the posterior, lack of convergence of the chain in the tail of the posterior, mixing problems, and misspecified priors on branch lengths. Here, we analyze the behavior of Bayesian Markov chain Monte Carlo algorithms when the chain is in the tail of the posterior distribution and note that all these phenomena can occur. In Bayesian phylogenetics, the likelihood function approaches a constant instead of zero when the branch lengths increase to infinity. The flat tail of the likelihood can cause poor mixing and undue influence of the prior. We suggest that the main cause of the extreme branch length estimates produced in many Bayesian analyses is the poor choice of a default prior on branch lengths in current Bayesian phylogenetic programs. The default prior in MrBayes assigns independent and identical distributions to branch lengths, imposing strong (and unreasonable) assumptions about the tree length. The problem is exacerbated by the strong correlation between the branch lengths and parameters in models of variable rates among sites or among site partitions. To resolve the problem, we suggest two multivariate priors for the branch lengths (called compound Dirichlet priors) that are fairly diffuse and demonstrate their utility in the special case of branch length estimation on a star phylogeny. Our analysis highlights the need for careful thought in the specification of high-dimensional priors in Bayesian analyses.     

The conflation of ignorance and knowledge in the inference of clade posteriors

The objective Bayesian approach relies on the construction of prior distributions that reflect ignorance. When topologies are considered equally probable a priori, clades cannot be. Shifting justifications have been offered for the use of uniform topological priors in Bayesian inference. These include: (i) topological priors do not inappropriately influence Bayesian inference when they are uniform; (ii) although clade priors are not uniform, their undesirable influence is negated by the likelihood function, even when data sets are small; and (iii) the influence of nonuniform clade priors is an appropriate reflection of knowledge. The first two justifications have been addressed previously: the first is false, and the second was found to be questionable. The third and most recent justification is inconsistent with the first two, and with the objective Bayesian philosophy itself. Thus, there has been no coherent justification for the use of nonflat clade priors in Bayesian phylogenetics. We discuss several solutions: (i) Bayesian inference can be abandoned in favour of other methods of phylogenetic inference; (ii) the objective Bayesian philosophy can be abandoned in favour of a subjective interpretation; (iii) the topology with the greatest posterior probability, which is also the tree of greatest marginal likelihood, can be accepted as optimal, with clade support estimated using other means; or (iv) a Bayes factor, which accounts for differences in priors among competing hypotheses, can be used to assess the weight of evidence in support of clades.
©The Willi Hennig Society 2009     

Quantification of prior impact in terms of effective current sample size

Bayesian methods allow borrowing of historical information through prior distributions. The concept of prior effective sample size (prior ESS) facilitates quantification and communication of such prior information by equating it to a sample size. Prior information can arise from historical observations; thus, the traditional approach identifies the ESS with such a historical sample size. However, this measure is independent of newly observed data, and thus would not capture an actual “loss of information” induced by the prior in case of prior-data conflict. We build on a recent work to relate prior impact to the number of (virtual) samples from the current data model and introduce the effective current sample size (ECSS) of a prior, tailored to the application in Bayesian clinical trial designs. Special emphasis is put on robust mixture, power, and commensurate priors. We apply the approach to an adaptive design in which the number of recruited patients is adjusted depending on the effective sample size at an interim analysis. We argue that the ECSS is the appropriate measure in this case, as the aim is to save current (as opposed to historical) patients from recruitment. Furthermore, the ECSS can help overcome lack of consensus in the ESS assessment of mixture priors and can, more broadly, provide further insights into the impact of priors. An R package accompanies the paper.     

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.     

Bayesian Semiparametric Models for Survival Data with a Cure Fraction

We propose methods for Bayesian inference for a new class of semiparametric survival models with a cure fraction. Specifically, we propose a semiparametric cure rate model with a smoothing parameter that controls the degree of parametricity in the right tail of the survival distribution. We show that such a parameter is crucial for these kinds of models and can have an impact on the posterior estimates. Several novel properties of the proposed model are derived. In addition, we propose a class of improper noninformative priors based on this model and examine the properties of the implied posterior. Also, a class of informative priors based on historical data is proposed and its theoretical properties are investigated. A case study involving a melanoma clinical trial is discussed in detail to demonstrate the proposed methodology.     

Bayesian analysis for generalized linear models with nonignorably missing covariates

We propose Bayesian methods for estimating parameters in generalized linear models (GLMs) with nonignorably missing covariate data. We show that when improper uniform priors are used for the regression coefficients, phi, of the multinomial selection model for the missing data mechanism, the resulting joint posterior will always be improper if (i) all missing covariates are discrete and an intercept is included in the selection model for the missing data mechanism, or (ii) at least one of the covariates is continuous and unbounded. This impropriety will result regardless of whether proper or improper priors are specified for the regression parameters, beta, of the GLM or the parameters, alpha, of the covariate distribution. To overcome this problem, we propose a novel class of proper priors for the regression coefficients, phi, in the selection model for the missing data mechanism. These priors are robust and computationally attractive in the sense that inferences about beta are not sensitive to the choice of the hyperparameters of the prior for phi and they facilitate a Gibbs sampling scheme that leads to accelerated convergence. In addition, we extend the model assessment criterion of Chen, Dey, and Ibrahim (2004a, Biometrika 91, 45-63), called the weighted L measure, to GLMs and missing data problems as well as extend the deviance information criterion (DIC) of Spiegelhalter et al. (2002, Journal of the Royal Statistical Society B 64, 583-639) for assessing whether the missing data mechanism is ignorable or nonignorable. A novel Markov chain Monte Carlo sampling algorithm is also developed for carrying out posterior computation. Several simulations are given to investigate the performance of the proposed Bayesian criteria as well as the sensitivity of the prior specification. Real datasets from a melanoma cancer clinical trial and a liver cancer study are presented to further illustrate the proposed methods.     

Fully Bayesian Inference for Structural MRI: Application to Segmentation and Statistical Analysis of T2-Hypointensities

Aiming at iron-related T2-hypointensity, which is related to normal aging and neurodegenerative processes, we here present two practicable approaches, based on Bayesian inference, for preprocessing and statistical analysis of a complex set of structural MRI data. In particular, Markov Chain Monte Carlo methods were used to simulate posterior distributions. First, we rendered a segmentation algorithm that uses outlier detection based on model checking techniques within a Bayesian mixture model. Second, we rendered an analytical tool comprising a Bayesian regression model with smoothness priors (in the form of Gaussian Markov random fields) mitigating the necessity to smooth data prior to statistical analysis. For validation, we used simulated data and MRI data of 27 healthy controls (age: ; range, ). We first observed robust segmentation of both simulated T2-hypointensities and gray-matter regions known to be T2-hypointense. Second, simulated data and images of segmented T2-hypointensity were analyzed. We found not only robust identification of simulated effects but also a biologically plausible age-related increase of T2-hypointensity primarily within the dentate nucleus but also within the globus pallidus, substantia nigra, and red nucleus. Our results indicate that fully Bayesian inference can successfully be applied for preprocessing and statistical analysis of structural MRI data.     

Effects of branch length uncertainty on Bayesian posterior probabilities for phylogenetic hypotheses

In Bayesian phylogenetics, confidence in evolutionary relationships is expressed as posterior probability--the probability that a tree or clade is true given the data, evolutionary model, and prior assumptions about model parameters. Model parameters, such as branch lengths, are never known in advance; Bayesian methods incorporate this uncertainty by integrating over a range of plausible values given an assumed prior probability distribution for each parameter. Little is known about the effects of integrating over branch length uncertainty on posterior probabilities when different priors are assumed. Here, we show that integrating over uncertainty using a wide range of typical prior assumptions strongly affects posterior probabilities, causing them to deviate from those that would be inferred if branch lengths were known in advance; only when there is no uncertainty to integrate over does the average posterior probability of a group of trees accurately predict the proportion of correct trees in the group. The pattern of branch lengths on the true tree determines whether integrating over uncertainty pushes posterior probabilities upward or downward. The magnitude of the effect depends on the specific prior distributions used and the length of the sequences analyzed. Under realistic conditions, however, even extraordinarily long sequences are not enough to prevent frequent inference of incorrect clades with strong support. We found that across a range of conditions, diffuse priors--either flat or exponential distributions with moderate to large means--provide more reliable inferences than small-mean exponential priors. An empirical Bayes approach that fixes branch lengths at their maximum likelihood estimates yields posterior probabilities that more closely match those that would be inferred if the true branch lengths were known in advance and reduces the rate of strongly supported false inferences compared with fully Bayesian integration.     

RECOMMENDATIONS FOR USING MSBAYES TO INCORPORATE UNCERTAINTY IN SELECTING AN ABC MODEL PRIOR: A RESPONSE TO OAKS ET AL.

Prior specification is an essential component of parameter estimation and model comparison in Approximate Bayesian computation (ABC). Oaks et al. present a simulation‐based power analysis of msBayes and conclude that msBayes has low power to detect genuinely random divergence times across taxa, and suggest the cause is Lindley's paradox. Although the predictions are similar, we show that their findings are more fundamentally explained by insufficient prior sampling that arises with poorly chosen wide priors that critically undersample nonsimultaneous divergence histories of high likelihood. In a reanalysis of their data on Philippine Island vertebrates, we show how this problem can be circumvented by expanding upon a previously developed procedure that accommodates uncertainty in prior selection using Bayesian model averaging. When these procedures are used, msBayes supports recent divergences without support for synchronous divergence in the Oaks et al. data and we further present a simulation analysis that demonstrates that msBayes can have high power to detect asynchronous divergence under narrower priors for divergence time. Our findings highlight the need for exploration of plausible parameter space and prior sampling efficiency for ABC samplers in high dimensions. We discus potential improvements to msBayes and conclude that when used appropriately with model averaging, msBayes remains an effective and powerful tool.     

Expert elicitation and data noise learning for material flow analysis using Bayesian inference

Bayesian inference allows the transparent communication and systematic updating of model uncertainty as new data become available. When applied to material flow analysis (MFA), however, Bayesian inference is undermined by the difficulty of defining proper priors for the MFA parameters and quantifying the noise in the collected data. We start to address these issues by first deriving and implementing an expert elicitation procedure suitable for generating MFA parameter priors. Second, we propose to learn the data noise concurrent with the parametric uncertainty. These methods are demonstrated using a case study on the 2012 US steel flow. Eight experts are interviewed to elicit distributions on steel flow uncertainty from raw materials to intermediate goods. The experts' distributions are combined and weighted according to the expertise demonstrated in response to seeding questions. These aggregated distributions form our model parameters' informative priors. Sensible, weakly informative priors are adopted for learning the data noise. Bayesian inference is then performed to update the parametric and data noise uncertainty given MFA data collected from the United States Geological Survey and the World Steel Association. The results show a reduction in MFA parametric uncertainty when incorporating the collected data. Only a modest reduction in data noise uncertainty was observed using 2012 data; however, greater reductions were achieved when using data from multiple years in the inference. These methods generate transparent MFA and data noise uncertainties learned from data rather than pre-assumed data noise levels, providing a more robust basis for decision-making that affects the system.     

Back to basics for Bayesian model building in genomic selection

Numerous Bayesian methods of phenotype prediction and genomic breeding value estimation based on multilocus association models have been proposed. Computationally the methods have been based either on Markov chain Monte Carlo or on faster maximum a posteriori estimation. The demand for more accurate and more efficient estimation has led to the rapid emergence of workable methods, unfortunately at the expense of well-defined principles for Bayesian model building. In this article we go back to the basics and build a Bayesian multilocus association model for quantitative and binary traits with carefully defined hierarchical parameterization of Student's t and Laplace priors. In this treatment we consider alternative model structures, using indicator variables and polygenic terms. We make the most of the conjugate analysis, enabled by the hierarchical formulation of the prior densities, by deriving the fully conditional posterior densities of the parameters and using the acquired known distributions in building fast generalized expectation-maximization estimation algorithms.