Approximate Bayesian Computation Methods Research Articles

Population Genetics and Effective Population Size of the Critically Endangered Nihoa Millerbird (Acrocephalus Familiaris Kingi)

ABSTRACT. Many species endemic to isolated islands are of conservation interest because of concerns over the potentially devastating effects of environmental stochasticity and the pending threat of anthropogenic influences and invasive species. The effective size (Ne) of these species is a key parameter in their conservation because it predicts the detrimental effects of inbreeding or genetic drift and can be used to inform management plans. We used microsatellite allele frequencies (4 loci) and mtDNA (control region; cytochrome b) sequences to assess the genetic diversity and the effective number of breeders (Nb) in the Nihoa Millerbird (Acrocephalus familiaris kingi), a critically endangered passerine endemic to the northwest Hawaiian island of Nihoa. Using samples collected in 2007 and 2009, our results reveal extremely low levels of genetic diversity at both microsatellites and mtDNA, and both approximate Bayesian computation (ABC) and sibship methods indicate that the effective number of breeders (Nb...

Approximate Bayesian Computation Using Indirect Inference

SummaryWe present a novel approach for developing summary statistics for use in approximate Bayesian computation (ABC) algorithms by using indirect inference. ABC methods are useful for posterior inference in the presence of an intractable likelihood function. In the indirect inference approach to ABC the parameters of an auxiliary model fitted to the data become the summary statistics. Although applicable to any ABC technique, we embed this approach within a sequential Monte Carlo algorithm that is completely adaptive and requires very little tuning. This methodological development was motivated by an application involving data on macroparasite population evolution modelled by a trivariate stochastic process for which there is no tractable likelihood function. The auxiliary model here is based on a beta–binomial distribution. The main objective of the analysis is to determine which parameters of the stochastic model are estimable from the observed data on mature parasite worms.

Incoherent phylogeographic inference

Templeton (1) makes a broad attack on the foundations of Bayesian statistical methods—rather than on the purely numerical technique called approximate Bayesian computation (ABC)—using incorrect arguments and selective references taken out of context. The most significant example is the argument, “The probability of the nested special case must be less than or equal to the probability of the general model within which the special case is nested. Any statistic that assigns greater probability to the special case is incoherent. An example of incoherence is shown in human evolution for ... the approximate Bayesian computation (ABC) method.” This opposes both the basis and the practice of Bayesian testing. The confusion seems to arise from misunderstanding the difference between scientific hypotheses and their mathematical representation. Consider vaccine testing, for which in what follows we use VE to represent the vaccine efficacy measured on a scale from −∞ to 100. Exploratory vaccines may be efficacious or not. Thus, a real biological model corresponds to the hypothesis “VE = 0,” that the vaccine is not efficacious. The alternative biological possibility, that the vaccine has an effect, is often stated mathematically as the alternative model “any allowed value of VE is possible,” making it seem that it contains “VE = 0.” However, Bayesian analysis assigns each model prior distributions arising from the background science; a point mass (e.g., probability ½) is assigned to “VE = 0,” and the remaining probability mass (e.g., ½) is distributed continuously over values of VE in the alternative model. Elementary use of Bayes’ theorem (see, e.g., ref. 2) then shows that the simpler model can indeed have a much higher posterior probability. Mathematically this is explained by the probability distributions residing in different dimensional spaces and is elementary probability theory for which use of Templeton's “Venn diagram argument” is simply incorrect. Templeton also argues that Bayes factors are mathematically incorrect, and he backs his claims with the notion of coherence of Lavine and Schervish (3). These authors do indeed criticize the use of Bayes factors as stand-alone criteria but point out that, when combined with prior probabilities of models (as illustrated in the vaccine example above), the result is fully coherent posterior probabilities. Furthermore, Templeton directly attacks the ABC algorithm. ABC is simply a numerical computational technique; attacking it as incoherent is similar to calling calculus incoherent if it is used to compute the wrong thing. Finally, we note that Templeton has already published essentially identical if more guarded arguments in the ecology literature; we refer readers to a related rebuttal to Templeton’s (4) critique of the Bayesian approach by Beaumont et al. (5) that is broader in scope, because it also covers the phylogenetic aspects of nested clade vs. a model-based approach.

Combining genetic, historical and geographical data to reconstruct the dynamics of bioinvasions: application to the cane toadBufo marinus

We developed a spatially explicit model of a bioinvasion and used an approximate Bayesian computation (ABC) framework to make various inferences from a combination of genetic (microsatellite genotypes), historical (first observation dates) and geographical (spatial coordinates of introduction and sampled sites) information. Our method aims to discriminate between alternative introduction scenarios and to estimate posterior densities of demographically relevant parameters of the invasive process. The performance of our landscape-ABC method is assessed using simulated data sets differing in their information content (genetic and/or historical data). We apply our methodology to the recent introduction and spatial expansion of the cane toad, Bufo marinus, in northern Australia. We find that, at least in the context of cane toad invasion, historical data are more informative than genetic data for discriminating between introduction scenarios. However, the combination of historical and genetic data provides the most accurate estimates of demographic parameters. For the cane toad, we find some evidence for a strong bottleneck prior to introduction, a small initial number of founder individuals (about 15), a large population growth rate (about 400% per generation), a standard deviation of dispersal distance of 19 km per generation and a high invasion speed at equilibrium (50 km per year). Our approach strengthens the application of the ABC method to the field of bioinvasion by allowing statistical inferences to be made on the introduction and the spatial expansion dynamics of invasive species using a combination of various relevant sources of information.

Two pulses of diversification across the Isthmus of Tehuantepec in a montane Mexican bird fauna

Understanding the evolutionary history of the species in a particular region provides insights into how that fauna was formed. Of particular interest to biogeographers is examining the impact a geographical barrier had in generating temporal genetic diversity among codistributed species. We examined the impact a major New World barrier, the Isthmus of Tehuantepec (IT) in southern Mexico, had on a regional bird fauna. Specifically, genetic data from 10 montane-forest bird taxa were analysed using approximate Bayesian computation (ABC) to test the hypothesis of simultaneous intraspecific diversification at the IT. Because effective population size (N(e)) has the greatest impact on coalescent times, thereby affecting tests of divergence among codistributed taxa, we chose priors for both current and ancestral N(e) using empirical estimates of theta. The ABC method detected two discrete diversification events. Subsequent analysis with the number of diversification events constrained to two suggests that four taxa diverged in an older event, with the remaining six diverging more recently. Application of a range of mutation rates from 2.0 to 5.0% Myr(-1) places both events within the Pleistocene or Late Pliocene, suggesting that fluctuations in montane habitat induced by climate cycles and a late Pliocene seaway may have fractured this montane bird fauna. The results presented here suggest this avian fauna responded in a relatively concerted fashion over the last several million years.

Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation

We estimate the parameters of a stochastic process model for a macroparasite population within a host using approximate Bayesian computation (ABC). The immunity of the host is an unobserved model variable and only mature macroparasites at sacrifice of the host are counted. With very limited data, process rates are inferred reasonably precisely. Modeling involves a three variable Markov process for which the observed data likelihood is computationally intractable. ABC methods are particularly useful when the likelihood is analytically or computationally intractable. The ABC algorithm we present is based on sequential Monte Carlo, is adaptive in nature, and overcomes some drawbacks of previous approaches to ABC. The algorithm is validated on a test example involving simulated data from an autologistic model before being used to infer parameters of the Markov process model for experimental data. The fitted model explains the observed extra-binomial variation in terms of a zero-one immunity variable, which has a short-lived presence in the host.

Coherent and incoherent inference in phylogeography and human evolution

A hypothesis is nested within a more general hypothesis when it is a special case of the more general hypothesis. Composite hypotheses consist of more than one component, and in many cases different composite hypotheses can share some but not all of these components and hence are overlapping. In statistics, coherent measures of fit of nested and overlapping composite hypotheses are technically those measures that are consistent with the constraints of formal logic. For example, the probability of the nested special case must be less than or equal to the probability of the general model within which the special case is nested. Any statistic that assigns greater probability to the special case is said to be incoherent. An example of incoherence is shown in human evolution, for which the approximate Bayesian computation (ABC) method assigned a probability to a model of human evolution that was a thousand-fold larger than a more general model within which the first model was fully nested. Possible causes of this incoherence are identified, and corrections and restrictions are suggested to make ABC and similar methods coherent. Another coalescent-based method, nested clade phylogeographic analysis, is coherent and also allows the testing of individual components of composite hypotheses, another attribute lacking in ABC and other coalescent-simulation approaches. Incoherence is a highly undesirable property because it means that the inference is mathematically incorrect and formally illogical, and the published incoherent inferences on human evolution that favor the out-of-Africa replacement hypothesis have no statistical or logical validity.

Bridgehead Effect in the Worldwide Invasion of the Biocontrol Harlequin Ladybird

Recent studies of the routes of worldwide introductions of alien organisms suggest that many widespread invasions could have stemmed not from the native range, but from a particularly successful invasive population, which serves as the source of colonists for remote new territories. We call here this phenomenon the invasive bridgehead effect. Evaluating the likelihood of such a scenario is heuristically challenging. We solved this problem by using approximate Bayesian computation methods to quantitatively compare complex invasion scenarios based on the analysis of population genetics (microsatellite variation) and historical (first observation dates) data. We applied this approach to the Harlequin ladybird Harmonia axyridis (HA), a coccinellid native to Asia that was repeatedly introduced as a biocontrol agent without becoming established for decades. We show that the recent burst of worldwide invasions of HA followed a bridgehead scenario, in which an invasive population in eastern North America acted as the source of the colonists that invaded the European, South American and African continents, with some admixture with a biocontrol strain in Europe. This demonstration of a mechanism of invasion via a bridgehead has important implications both for invasion theory (i.e., a single evolutionary shift in the bridgehead population versus multiple changes in case of introduced populations becoming invasive independently) and for ongoing efforts to manage invasions by alien organisms (i.e., heightened vigilance against invasive bridgeheads).

Parameter inference in small world network disease models with approximate Bayesian Computational methods

Small world network models have been effective in capturing the variable behaviour of reported case data of the SARS coronavirus outbreak in Hong Kong during 2003. Simulations of these models have previously been realized using informed “guesses” of the proposed model parameters and tested for consistency with the reported data by surrogate analysis. In this paper we attempt to provide statistically rigorous parameter distributions using Approximate Bayesian Computation sampling methods. We find that such sampling schemes are a useful framework for fitting parameters of stochastic small world network models where simulation of the system is straightforward but expressing a likelihood is cumbersome.

Comparative phylogeography among hydrothermal vent species along the East Pacific Rise reveals vicariant processes and population expansion in the South

The use of sequence polymorphism from individual mitochondrial genes to infer past demography has recently proved controversial because of the recurrence of selective sweeps acting over genes and the need for unlinked multilocus data sets. However, comparative analyses using several species for one gene and/or multiple genes for one species can serve as a test for potential selective effects and clarify our understanding of historical demographic effects. This study compares nucleotide polymorphisms in mitochondrial cytochrome oxidase I across seven deep-sea hydrothermal vent species that live along the volcanically active East Pacific Rise. Approximate Bayesian Computation (ABC) method, developed to trace shared vicariant events across species pairs, indicates the occurrence of two across species divergence times, and suggests that the present geographical patterns of genetic differentiation may be explained by two periods of significant population isolation. The oldest period dates back 11.6 Ma and is associated with the vent limpet Lepetodrilus elevatus, while the most recent period of isolation is 1.3 Ma, which apparently affected all species examined and coincides with a transition zone across the equator. Moreover, significant negative Tajima's D and star-like networks were observed for all southern lineages, suggesting that these lineages experienced a concomitant demographic and geographical expansion about 100 000-300 000 generations ago. This expansion may have initiated from a wave of range expansions during the secondary colonization of new sites along the Southern East Pacific Rise (founder effects below the equator) or recurrent bottleneck events because of the increase of eruptive phases associated with the higher spreading rates of the ridge in this region.

Inferring introduction routes of invasive species using approximate Bayesian computation on microsatellite data

Determining the routes of introduction provides not only information about the history of an invasion process, but also information about the origin and construction of the genetic composition of the invading population. It remains difficult, however, to infer introduction routes from molecular data because of a lack of appropriate methods. We evaluate here the use of an approximate Bayesian computation (ABC) method for estimating the probabilities of introduction routes of invasive populations based on microsatellite data. We considered the crucial case of a single source population from which two invasive populations originated either serially from a single introduction event or from two independent introduction events. Using simulated datasets, we found that the method gave correct inferences and was robust to many erroneous beliefs. The method was also more efficient than traditional methods based on raw values of statistics such as assignment likelihood or pairwise F(ST). We illustrate some of the features of our ABC method, using real microsatellite datasets obtained for invasive populations of the western corn rootworm, Diabrotica virgifera virgifera. Most computations were performed with the DIYABC program (http://www1.montpellier.inra.fr/CBGP/diyabc/).

Approximate Bayesian Computation Without Summary Statistics: The Case of Admixture

In recent years approximate Bayesian computation (ABC) methods have become popular in population genetics as an alternative to full-likelihood methods to make inferences under complex demographic models. Most ABC methods rely on the choice of a set of summary statistics to extract information from the data. In this article we tested the use of the full allelic distribution directly in an ABC framework. Although the ABC techniques are becoming more widely used, there is still uncertainty over how they perform in comparison with full-likelihood methods. We thus conducted a simulation study and provide a detailed examination of ABC in comparison with full likelihood in the case of a model of admixture. This model assumes that two parental populations mixed at a certain time in the past, creating a hybrid population, and that the three populations then evolve under pure drift. Several aspects of ABC methodology were investigated, such as the effect of the distance metric chosen to measure the similarity between simulated and observed data sets. Results show that in general ABC provides good approximations to the posterior distributions obtained with the full-likelihood method. This suggests that it is possible to apply ABC using allele frequencies to make inferences in cases where it is difficult to select a set of suitable summary statistics and when the complexity of the model or the size of the data set makes it computationally prohibitive to use full-likelihood methods.

Inferring the parameters of the neutral theory of biodiversity using phylogenetic information and implications for tropical forests

We develop a statistical method to infer the parameters of Hubbell's neutral model of biodiversity using data on local species abundances and their phylogenetic relatedness. This method uses the approximate Bayesian computation (ABC) approach, where the data are summarized into a small number of informative summary statistics. We used three statistics: the number of species in the sample, Shannon H index of evenness and Shao and Sokal's B(1) index of phylogenetic tree imbalance. Our approach was found to outperform previous methods, illustrating the potential of ABC methods in ecology. Applying it to four large tropical forest tree data sets, the best-fit immigration rates m were found to be two orders of magnitude smaller and regional diversities theta larger than previously reported for the same data. This implies that neutral-compatible regional pools of tropical trees should extend over continental scales, and that m measures, in this context, mostly the frequency of long-distance dispersal events.

Origins and Genetic Diversity of Pygmy Hunter-Gatherers from Western Central Africa

Central Africa is currently peopled by numerous sedentary agriculturalist populations neighboring the largest group of mobile hunter-gatherers, the Pygmies [1-3]. Although archeological remains attest to Homo sapiens' presence in the Congo Basin for at least 30,000 years, the demographic history of these groups, including divergence and admixture, remains widely unknown [4-6]. Moreover, it is still debated whether common history or convergent adaptation to a forest environment resulted in the short stature characterizing the pygmies [2, 7]. We genotyped 604 individuals at 28 autosomal tetranucleotide microsatellite loci in 12 nonpygmy and 9 neighboring pygmy populations. We found a high level of genetic heterogeneity among Western Central African pygmies, as well as evidence of heterogeneous levels of asymmetrical gene flow from nonpygmies to pygmies, consistent with the variable sociocultural barriers against intermarriages. Using approximate Bayesian computation (ABC) methods [8], we compared several historical scenarios. The most likely points toward a unique ancestral pygmy population that diversified approximately 2800 years ago, contemporarily with the Neolithic expansion of nonpygmy agriculturalists [9, 10]. Our results show that recent isolation, genetic drift, and heterogeneous admixture enabled a rapid and substantial genetic differentiation among Western Central African pygmies. Such an admixture pattern is consistent with the various sociocultural behaviors related to intermariages between pygmies and nonpygmies.

Rapid radiation in spiny lobsters (Palinurus spp) as revealed by classic and ABC methods using mtDNA and microsatellite data

BackgroundMolecular tools may help to uncover closely related and still diverging species from a wide variety of taxa and provide insight into the mechanisms, pace and geography of marine speciation. There is a certain controversy on the phylogeography and speciation modes of species-groups with an Eastern Atlantic-Western Indian Ocean distribution, with previous studies suggesting that older events (Miocene) and/or more recent (Pleistocene) oceanographic processes could have influenced the phylogeny of marine taxa. The spiny lobster genus Palinurus allows for testing among speciation hypotheses, since it has a particular distribution with two groups of three species each in the Northeastern Atlantic (P. elephas, P. mauritanicus and P. charlestoni) and Southeastern Atlantic and Southwestern Indian Oceans (P. gilchristi, P. delagoae and P. barbarae). In the present study, we obtain a more complete understanding of the phylogenetic relationships among these species through a combined dataset with both nuclear and mitochondrial markers, by testing alternative hypotheses on both the mutation rate and tree topology under the recently developed approximate Bayesian computation (ABC) methods.ResultsOur analyses support a North-to-South speciation pattern in Palinurus with all the South-African species forming a monophyletic clade nested within the Northern Hemisphere species. Coalescent-based ABC methods allowed us to reject the previously proposed hypothesis of a Middle Miocene speciation event related with the closure of the Tethyan Seaway. Instead, divergence times obtained for Palinurus species using the combined mtDNA-microsatellite dataset and standard mutation rates for mtDNA agree with known glaciation-related processes occurring during the last 2 my.ConclusionThe Palinurus speciation pattern is a typical example of a series of rapid speciation events occurring within a group, with very short branches separating different species. Our results support the hypothesis that recent climate change-related oceanographic processes have influenced the phylogeny of marine taxa, with most Palinurus species originating during the last two million years. The present study highlights the value of new coalescent-based statistical methods such as ABC for testing different speciation hypotheses using molecular data.

Genetic structure of the northwestern Russian wolf populations and gene flow between Russia and Finland

We examined the genetic diversity and structure of wolf populations in northwestern Russia. Populations in Republic of Karelia and Arkhangelsk Oblast were sampled during 1995–2000, and 43 individuals were genotyped with 10 microsatellite markers. Moreover, 118 previously genotyped wolves from the neighbouring Finnish population were used as a reference population. A relatively large amount of genetic variation was found in the Russian populations, and the Karelian wolf population tended to be slightly more polymorphic than the Arkhangelsk population. We found significant inbreeding (F = 0.094) in the Karelian, but not in the Arkhangelsk population. The effective size estimates of the Karelian wolf population based on the approximate Bayesian computation and linkage disequilibrium methods were 39.9 and 46.7 individuals, respectively. AMOVA-analysis and exact test of population differentiation suggested clear differentiation between the Karelian, Arkhangelsk and Finnish wolf populations. Indirect estimates of gene flow based on the level of population differentiation (ϕST = 0.152) and frequency of private alleles (0.029) both suggested a low level of gene flow between the populations (Nm = 1.4 and Nm = 3.7, respectively). Assignment analysis of Karelian and Finnish populations suggested an even lower number of recent migrants (less than 0.03) between populations, with a larger amount of migration from Finland to Karelia than vice versa. Our findings emphasise the role of physical obstacles and territorial behaviour in creating barriers to gene flow between populations in relatively limited geographical areas, even in large-bodied mammalian species with long-distance dispersal capabilities and an apparently continuous population structure.

Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems

Approximate Bayesian computation (ABC) methods can be used to evaluate posterior distributions without having to calculate likelihoods. In this paper, we discuss and apply an ABC method based on sequential Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC SMC provides information about the inferability of parameters and model sensitivity to changes in parameters, and tends to perform better than other ABC approaches. The algorithm is applied to several well-known biological systems, for which parameters and their credible intervals are inferred. Moreover, we develop ABC SMC as a tool for model selection; given a range of different mathematical descriptions, ABC SMC is able to choose the best model using the standard Bayesian model selection apparatus.

Introduction history of Drosophila subobscura in the New World: a microsatellite‐based survey using ABC methods

Drosophila subobscura is a Palearctic species that was first observed in South and North America in the early 1980s, and that rapidly invaded broad latitudinal ranges on both continents. To trace the source and history of this invasion, we obtained genotypic data on nine microsatellite loci from two South American, two North American and five European populations of D. subobscura. We analysed these data with traditional statistics as well as with an approximate Bayesian computation (ABC) framework. ABC methods yielded the strongest support for the scenario involving a serial introduction with founder events from Europe into South America, and then from South America into North America. Stable effective population size of the source population was very large (around one million individuals), and the propagule size was notably smaller for the introduction into South America (i.e. high bottleneck severity index with only a few effective founders) but considerably larger for the subsequent introduction into North America (i.e. low bottleneck severity index with around 100-150 effective founders). Finally, the Mediterranean region of Europe (and most likely Barcelona from the localities so far analysed) is proposed as the source of the New World flies, based on mean individual assignment statistics.