Bayesian Computation Methods Research Articles

Generative Bayesian Computation for Maximum Expected Utility.

Generative Bayesian Computation (GBC) methods are developed to provide an efficient computational solution for maximum expected utility (MEU). We propose a density-free generative method based on quantiles that naturally calculates expected utility as a marginal of posterior quantiles. Our approach uses a deep quantile neural estimator to directly simulate distributional utilities. Generative methods only assume the ability to simulate from the model and parameters and as such are likelihood-free. A large training dataset is generated from parameters, data and a base distribution. Then, a supervised learning problem is solved as a non-parametric regression of generative utilities on outputs and base distribution. We propose the use of deep quantile neural networks. Our method has a number of computational advantages, primarily being density-free and an efficient estimator of expected utility. A link with the dual theory of expected utility and risk taking is also described. To illustrate our methodology, we solve an optimal portfolio allocation problem with Bayesian learning and power utility (also known as the fractional Kelly criterion). Finally, we conclude with directions for future research.

Evaluation of Tolerance Selection Strategies and Multifidelity Techniques in ABC Methods

Approximate Bayesian Computation (ABC) methods provides a flexible and robust framework for solve model fitting problems, particularly pertinent to complex models with intractable likelihood functions. This methodology is based on approximating a simulated parameter values through auxiliary data and evaluating the distance of this data with the true dataset. Effective implementation of these methods require reasonable decisions to be made about the selection of ABC techniques and how to implement the algorithm to ensure computational efficiency. A comprehensive understanding of these factors facilitates a more accurate inference about the model. This study aims to explore the impact of the tolerance selection and the integration of Multifidelity techniques to the convergence and cost of the method. In the initial approach, three methods of choosing tolerances are employed: a predefined vector, a percentile calculation, and a percentage calculation based on the distance vector obtained from model simulations, in order to verify the most effective strategy for the improvement of the inference process. Subsequently, the optimal approach of tolerances selection is combined with Multifidelity techniques to reduce the computational cost. This methodology is applied in the examination of infectious disease models of differential equations. This study aims to highlight the importance of careful decision-making when using ABC methods, through the evaluation of some approaches for choosing tolerance in combination with techniques that aim to reduce the computational cost in constructing samples in ABC methods. Through this analysis, we focus our efforts on trying to demonstrate insights about gains in computational efficiency and accuracy in results.

Harnessing 12-lead ECG and MRI data to personalise repolarisation profiles in cardiac digital twin models for enhanced virtual drug testing

Cardiac digital twins are computational tools capturing key functional and anatomical characteristics of patient hearts for investigating disease phenotypes and predicting responses to therapy. When paired with large-scale computational resources and large clinical datasets, digital twin technology can enable virtual clinical trials on virtual cohorts to fast-track therapy development. Here, we present an open-source automated pipeline for personalising ventricular electrophysiological function based on routinely acquired magnetic resonance imaging (MRI) data and the standard 12-lead electrocardiogram (ECG).Using MRI-based anatomical models, a sequential Monte-Carlo approximate Bayesian computational inference method is extended to infer electrical activation and repolarisation characteristics from the ECG. Fast simulations are conducted with a reaction-Eikonal model, including the Purkinje network and biophysically-detailed subcellular ionic current dynamics for repolarisation. For each patient, parameter uncertainty is represented by inferring an envelope of plausible ventricular models rather than a single one, which means that parameter uncertainty can be propagated to therapy evaluation. Furthermore, we have developed techniques for translating from reaction-Eikonal to monodomain simulations, which allows more realistic simulations of cardiac electrophysiology. The pipeline is demonstrated in three healthy subjects, where our inferred pseudo-diffusion reaction-Eikonal models reproduced the patient's ECG with a median Pearson's correlation coefficient of 0.9, and then translated to monodomain simulations with a median correlation coefficient of 0.84 across all subjects. We then demonstrate our digital twins for virtual evaluation of Dofetilide with uncertainty quantification. These evaluations using our cardiac digital twins reproduced dose-dependent QTc and T peak to T end prolongations that are in keeping with large population drug response data.The methodologies for cardiac digital twinning presented here are a step towards personalised virtual therapy testing and can be scaled to generate virtual populations for clinical trials to fast-track therapy evaluation. The tools developed for this paper are open-source, documented, and made publicly available.

On an Empirical Likelihood Based Solution to the Approximate Bayesian Computation Problem

ABSTRACTApproximate Bayesian computation (ABC) methods are applicable to statistical models specified by generative processes with analytically intractable likelihoods. These methods try to approximate the posterior density of a model parameter by comparing the observed data with additional process‐generated simulated data sets. For computational benefit, only the values of certain well‐chosen summary statistics are usually compared, instead of the whole data set. Most ABC procedures are computationally expensive, justified only heuristically, and have poor asymptotic properties. In this article, we introduce a new empirical likelihood‐based approach to the ABC paradigm called ABCel. The proposed procedure is computationally tractable and approximates the target log posterior of the parameter as a sum of two functions of the data—namely, the mean of the optimal log‐empirical likelihood weights and the estimated differential entropy of the summary functions. We rigorously justify the procedure via direct and reverse information projections onto appropriate classes of probability densities. Past applications of empirical likelihood in ABC demanded constraints based on analytically tractable estimating functions that involve both the data and the parameter; although by the nature of the ABC problem such functions may not be available in general. In contrast, we use constraints that are functions of the summary statistics only. Equally importantly, we show that our construction directly connects to the reverse information projection and estimate the relevant differential entropy by a k‐NN estimator. We show that ABCel is posterior consistent and has highly favorable asymptotic properties. Its construction justifies the use of simple summary statistics like moments, quantiles, and so forth, which in practice produce accurate approximation of the posterior density. We illustrate the performance of the proposed procedure in a range of applications.

Inference of host-pathogen interaction matrices from genome-wide polymorphism data.

Host-pathogen coevolution is defined as the reciprocal evolutionary changes in both species due to genotype x genotype (GxG) interactions at the genetic level determining the outcome and severity of infection. While co-analyses of host and pathogen genomes (co-GWAs) allow us to pinpoint the interacting genes, these do not reveal which host genotype(s) is/are resistant to which pathogen genotype(s). The knowledge of this so-called infection matrix is important for agriculture and medicine. Building on established theories of host-pathogen interactions, we here derive four novel indices capturing the characteristics of the infection matrix. These indices can be computed from full genome polymorphism data of randomly sampled uninfected hosts, as well as infected hosts and their pathogen strains. We use these indices in an Approximate Bayesian Computation method to pinpoint loci with relevant GxG interactions and to infer their underlying interaction matrix. In a combined SNP data set of 451 European humans and their infecting Hepatitis C Virus (HCV) strains and 503 uninfected individuals, we reveal a new human candidate gene for resistance to HCV and new virus mutations matching human genes. For two groups of significant human-HCV (GxG) associations, we infer a gene-for-gene infection matrix, which is commonly assumed to be typical of plant-pathogen interactions. Our model-based inference framework bridges theoretical models of GxG interactions with host and pathogen genomic data. It, therefore, paves the way for understanding the evolution of key GxG interactions underpinning HCV adaptation to the European human population after a recent expansion.

Global transcription regulation revealed from dynamical correlations in time-resolved single-cell RNA sequencing

Single-cell transcriptomics reveals significant variations in transcriptional activity across cells. Yet, it remains challenging to identify mechanisms of transcription dynamics from static snapshots. It is thus still unknown what drives global transcription dynamics in single cells. We present a stochastic model of gene expression with cell size- and cell cycle-dependent rates in growing and dividing cells that harnesses temporal dimensions of single-cell RNA sequencing through metabolic labeling protocols and cel lcycle reporters. We develop a parallel and highly scalable approximate Bayesian computation method that corrects for technical variation and accurately quantifies absolute burst frequency, burst size, and degradation rate along the cell cycle at a transcriptome-wide scale. Using Bayesian model selection, we reveal scaling between transcription rates and cell size and unveil waves of gene regulation across the cell cycle-dependent transcriptome. Our study shows that stochastic modeling of dynamical correlations identifies global mechanisms of transcription regulation. A record of this paper's transparent peer review process is included in the supplemental information.

Mathematical Modelling of Parasite Dynamics: A Stochastic Simulation-Based Approach and Parameter Estimation via Modified Sequential-Type Approximate Bayesian Computation

The development of mathematical models for studying newly emerging and re-emerging infectious diseases has gained momentum due to global events. The gyrodactylid-fish system, like many host-parasite systems, serves as a valuable resource for ecological, evolutionary, and epidemiological investigations owing to its ease of experimental manipulation and long-term monitoring. Although this system has an existing individual-based model, it falls short in capturing information about species-specific microhabitat preferences and other biological details for different Gyrodactylus strains across diverse fish populations. This current study introduces a new individual-based stochastic simulation model that uses a hybrid τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-leaping algorithm to incorporate this essential data, enhancing our understanding of the complexity of the gyrodactylid-fish system. We compare the infection dynamics of three gyrodactylid strains across three host populations. A modified sequential-type approximate Bayesian computation (ABC) method, based on sequential Monte Carlo and sequential importance sampling, is developed. Additionally, we establish two penalised local-linear regression methods (based on L1 and L2 regularisations) for ABC post-processing analysis to fit our model using existing empirical data. With the support of experimental data and the fitted mathematical model, we address open biological questions for the first time and propose directions for future studies on the gyrodactylid-fish system. The adaptability of the mathematical model extends beyond the gyrodactylid-fish system to other host-parasite systems. Furthermore, the modified ABC methodologies provide efficient calibration for other multi-parameter models characterised by a large set of correlated or independent summary statistics.

Impact of time resolution on estimation of energy savings using a copula-based calibration in UBEM

Urban Building Energy modelling (UBEM) has emerged in the last decade as an important tool to accelerate energy transition in the building sector. To fulfil one of its major purposes - forecasting energy savings for potential energy conservation measures at an urban scale, several challenges are still to be addressed. Two key challenges include: 1) the need to calibrate models with a large number of unknown parameters across numerous buildings (either individually or through archetype representation); 2) the limited availability of high-resolution measured data, which raises concerns about calibrated models based solely on yearly values for accurate energy savings forecast. This study addresses these challenges using a case study of 35 buildings in a district in Stockholm, Sweden. Firstly, a new iterative Approximate Bayesian Computation (ABC) method for calibration is proposed, incorporating a copula-based sampling process at each iteration. Secondly, the new calibration process is applied with three different time resolutions to examine the impact of data resolution on forecasted energy usage, particularly when applied to a classical energy conservation measure. Results demonstrate the efficiency of the new method across the 35 buildings, facilitating the rapid population of a final joint distribution for the nine unknown parameters considered in each building. While the marginals may be strongly influenced by the time resolution, the forecasted energy consumption remains identical across the three analysed time resolutions. However, a noticeable difference is observed when the ECM pertains to a formerly unknown or known parameter.

Data-driven Stochastic Model for Quantifying the Interplay Between Amyloid-beta and Calcium Levels in Alzheimer's Disease.

The abnormal aggregation of extracellular amyloid- in senile plaques resulting in calcium dyshomeostasis is one of the primary symptoms of Alzheimer's disease (AD). Significant research efforts have been devoted in the past to better understand the underlying molecular mechanisms driving deposition and dysregulation. Importantly, synaptic impairments, neuronal loss, and cognitive failure in AD patients are all related to the buildup of intraneuronal accumulation. Moreover, increasing evidence show a feed-forward loop between and levels, i.e. disrupts neuronal levels, which in turn affects the formation of . To better understand this interaction, we report a novel stochastic model where we analyze the positive feedback loop between and using ADNI data. A good therapeutic treatment plan for AD requires precise predictions. Stochastic models offer an appropriate framework for modelling AD since AD studies are observational in nature and involve regular patient visits. The etiology of AD may be described as a multi-state disease process using the approximate Bayesian computation method. So, utilizing ADNI data from 2-year visits for AD patients, we employ this method to investigate the interplay between and levels at various disease development phases. Incorporating the ADNI data in our physics-based Bayesian model, we discovered that a sufficiently large disruption in either metabolism or intracellular homeostasis causes the relative growth rate in both and , which corresponds to the development of AD. The imbalance of ions causes disorders by directly or indirectly affecting a variety of cellular and subcellular processes, and the altered homeostasis may worsen the abnormalities of ion transportation and deposition. This suggests that altering the balance or the balance between and by chelating them may be able to reduce disorders associated with AD and open up new research possibilities for AD therapy.

A MCMC method based on surrogate model and Gaussian process parameterization for infinite Bayesian PDE inversion

This work focuses on an inversion problem derived from parametric partial differential equations (PDEs) with an infinite-dimensional parameter, represented as a coefficient function. The objective is to estimate this coefficient function accurately despite having only noisy measurements of the PDE solution at sparse input points. Conventional methods for inversion require numerous calls to a refined PDE solver, resulting in significant computational complexity, especially for challenging PDE problems, making the inference of the coefficient function practically infeasible. To address this issue, we propose a MCMC method that combines an deep learning-based surrogate and Gaussian process parameterization to efficiently infer the posterior of the coefficient function. The surrogate model is a combination of a cost-effective coarse PDE solver and a neural network-based transformation which provides an approximate solution derived from the refined PDE solver but based on the coarse PDE solution. The coefficient function is represented by Gaussian process with finite number of spatially dependent parameters and this parametric representation will be beneficial for the preconditioned Crank-Nicolson (pCN) Markov chain Monte Carlo method to sample efficiently from the posterior distribution. Approximate Bayesian computation method is used for constructing an informative dataset for the transformation learning. Our numerical examples demonstrate the effectiveness of this approach in accurately estimating the coefficient function while significantly reducing the computational burden associated with traditional inversion techniques.

Probabilistic Offline Policy Ranking with Approximate Bayesian Computation

In practice, it is essential to compare and rank candidate policies offline before real-world deployment for safety and reliability. Prior work seeks to solve this offline policy ranking (OPR) problem through value-based methods, such as Off-policy evaluation (OPE). However, they fail to analyze special case performance (e.g., worst or best cases), due to the lack of holistic characterization of policies’ performance. It is even more difficult to estimate precise policy values when the reward is not fully accessible under sparse settings. In this paper, we present Probabilistic Offline Policy Ranking (POPR), a framework to address OPR problems by leveraging expert data to characterize the probability of a candidate policy behaving like experts, and approximating its entire performance posterior distribution to help with ranking. POPR does not rely on value estimation, and the derived performance posterior can be used to distinguish candidates in worst-, best-, and average-cases. To estimate the posterior, we propose POPR-EABC, an Energy-based Approximate Bayesian Computation (ABC) method conducting likelihood-free inference. POPR-EABC reduces the heuristic nature of ABC by a smooth energy function, and improves the sampling efficiency by a pseudo-likelihood. We empirically demonstrate that POPR-EABC is adequate for evaluating policies in both discrete and continuous action spaces across various experiment environments, and facilitates probabilistic comparisons of candidate policies before deployment.

Approximate Bayesian computational methods to estimate the strength of divergent selection in population genomics models

Statistical estimation of parameters in large models of evolutionary processes is often too computationally inefficient to pursue using exact model likelihoods, even with single-nucleotide polymorphism (SNP) data, which offers a way to reduce the size of genetic data while retaining relevant information. Approximate Bayesian Computation (ABC) to perform statistical inference about parameters of large models takes the advantage of simulations to bypass direct evaluation of model likelihoods. We develop a mechanistic model to simulate forward-in-time divergent selection with variable migration rates, modes of reproduction (sexual, asexual), length and number of migration-selection cycles. We investigate the computational feasibility of ABC to perform statistical inference and study the quality of estimates on the position of loci under selection and the strength of selection. To expand the parameter space of positions under selection, we enhance the model by implementing an outlier scan on summarized observed data. We evaluate the usefulness of summary statistics well-known to capture the strength of selection, and assess their informativeness under divergent selection. We also evaluate the effect of genetic drift with respect to an idealized deterministic model with single-locus selection. We discuss the role of the recombination rate as a confounding factor in estimating the strength of divergent selection, and emphasize its importance in break down of linkage disequilibrium (LD). We answer the question for which part of the parameter space of the model we recover strong signal for estimating the selection, and determine whether population differentiation-based summary statistics or LD–based summary statistics perform well in estimating selection.

Quantifying health risks from ESBL-producing Escherichia coli in Dutch broiler production chains and potential interventions using compartmental models

Extended-spectrum beta-lactamase (ESBL)-producing Escherichia coli (E. coli) in animals are considered a human health threat, because this type of bacteria can serve as a reservoir of antibiotic resistant genes and act as a continuous threat of the emergence of new resistant bacteria, in addition to the direct effect of making infection untreatable. Although the prevalence of ESBL producing bacteria in broilers was drastically reduced in the Netherlands, chicken meat still has the highest prevalence among meat products. Therefore, further control of the ESBL-producing E. coli in the broiler production chain is important to reduce public health risks. The main objectives of this study were to evaluate the effectiveness of intervention scenarios to reduce the transmission of ESBL-producing E. coli in the broiler production chain and to quantitatively estimate the risk to public health. In this study, we developed two different types of transmission models that described the observed time-related decline in prevalence during a production round: one with time-dependent decline in susceptibility and one with partial immunity to phylogenetic groups. Both models incorporated the environmental contamination effect between production rounds and within flocks. The parameter values, including transmission rate and recovery rate, were estimated by Approximate Bayesian computation (ABC) method using data from a longitudinal study in a Dutch organic broiler farm. We applied the models to the three production stages in the broiler production chain, beginning from the Parent Stock (PS) farms, the hatcheries, and to the broiler farms. In our models, eggs were collected from different parent stock farms and transported to the hatchery and from there to a broiler farm.The size of a flock and the number of farms were adjusted to the Dutch situation. Both models were able to describe the observed dynamics within and between the production stages equally well, with estimated ESBL-producing E. coli prevalence of 8.98% and 11.47% in broilers at slaughter and 0.12% and 0.15% in humans due to chicken consumption. Both models indicated that improving farm management to eliminate the bacteria from the environment was the most effective intervention, making this outcome robust. Although chicken meat consumption is not a major risk factor for human carriage of the bacteria according to our models, reducing the bacteria in the PS and broiler farm environment to at least one percent can further decrease the prevalence in humans.

Selection of models and parameter estimation for monitoring the COVID-19 epidemic in Brazil via Bayesian inference

In 2019, a new strain of coronavirus led to an outbreak of disease cases named COVID-19, evolving rapidly into a pandemic. In Brazil, delayed decision making and lack of knowledge have resulted in an alarming increase in daily transmission and deaths. In this context, researchers used mathematical models to assist in determining the parameters that act in the spread of diseases, revealing containment measures. However, numerous mathematical models exist in the literature, each with specific parameters to be specified, leading to an important question about which model best represents the pandemic behavior. In this regard, this work aims to apply the Approximate Bayesian Computation method to select the best model and simultaneously estimate the parameters to resolve the abovementioned issue. The models adopted were susceptible-infected-recovered (SIR), susceptible-exposed-infected-recovered (SEIR), susceptible-infected-recovered-susceptible (SIRS), and susceptible-exposed-infected-recovered-susceptible (SEIRS). Approximate Bayesian Computation Monte Carlo Sequencing (ABC-SMC) was used to select among four competing models to represent the number of infected individuals and to estimate the model parameters based on three periods of Brazil COVID-19 data. A forecasting test was performed to test the ABC-SMC algorithm and the selected models for two months. The result was compared with the actual number of infected that were reported. Among the teste models, it was found that the ABC-SMC algorithm had a promising performance, since the data were noisy and the models could not predict all parameters.

Predicting clinical decline over 36 months with a memory‐based digital biomarker

AbstractBackgroundWord‐list recall tests are routinely employed in clinical practice for dementia screening. Process scoring and latent modeling of these tests, where underlying neurocognitive mechanisms are exploited, have shown potential to enhance test accuracy without requiring test redesign. We examined Embic’s digital biomarker M, a measure of recall derived from multinomial processing trees and hierarchical Bayesian computational methods, as a predictor of conversion to mild cognitive impairment (MCI) and Alzheimer’s disease (AD), and of increase in Clinical Dementia Rating (CDR) score.MethodsSecondary data analyses were carried out from ADNI data. M was computed from Rey’s AVLT data, and compared to standard AVLT clinical metrics (total and delayed recall). We conducted logistic regression analyses with diagnosis at 36 months (normal vs. MCI + AD) as outcome; all 169 participants were classified as normal at baseline, and 13 of these progressed to a clinical diagnosis (see Table 1). We controlled for baseline age, gender and years of education; and ran separate analyses with other metrics to avoid multicollinearity. We then repeated the same analytical framework with CDR scores (stable at 0 vs. increased to 0.5 or 1) at 36 months as outcome: stable participants were 141, and 24 increased (see Table 2). Covariates and predictors were the same as in the previous analysis.Results M predicted conversion to MCI or AD after three years (FDR‐corrected p value = 0.048), while neither AVLT total nor delayed recall did. Performance metrics showed that M generated 92% accuracy, or probability of yielding a correct classification, and 100% specificity, or no false positives (Figure 1). Analogously, M, but not standard AVLT metrics, also predicted a CDR increase after three years (FDR‐corrected p value = 0.003), with high accuracy (84%), specificity (96%), and precision (80%), or positive predictive value (Figure 2). Sensitivity analyses with ordinal regression broadly confirmed these findings.ConclusionsThese results indicate Embic’s digital biomarker M outperforms some of AVLT’s traditional metrics in identifying individuals who will progress to MCI and AD after three years, from a healthy baseline.

Parameter estimation for fractional power type diffusion: A hybrid Bayesian-deep learning approach

. In this article, we consider the problem of parameter estimation in a power-type diffusion driven by fractional Brownian motion with Hurst parameter in ( 1 / 2 , 1 ) . To estimate the parameters of the process, we use an approximate bayesian computation method. Also, a particular case is addressed by means of variations and wavelet-type methods. Several theoretical properties of the process are studied and numerical examples are provided in order to show the small sample behavior of the proposed methods.

Laplace based Bayesian inference for ordinary differential equation models using regularized artificial neural networks

Parameter estimation and associated uncertainty quantification is an important problem in dynamical systems characterised by ordinary differential equation (ODE) models that are often nonlinear. Typically, such models have analytically intractable trajectories which result in likelihoods and posterior distributions that are similarly intractable. Bayesian inference for ODE systems via simulation methods require numerical approximations to produce inference with high accuracy at a cost of heavy computational power and slow convergence. At the same time, Artificial Neural Networks (ANN) offer tractability that can be utilized to construct an approximate but tractable likelihood and posterior distribution. In this paper we propose a hybrid approach, where Laplace-based Bayesian inference is combined with an ANN architecture for obtaining approximations to the ODE trajectories as a function of the unknown initial values and system parameters. Suitable choices of customized loss functions are proposed to fine tune the approximated ODE trajectories and the subsequent Laplace approximation procedure. The effectiveness of our proposed methods is demonstrated using an epidemiological system with non-analytical solutions—the Susceptible-Infectious-Removed (SIR) model for infectious diseases—based on simulated and real-life influenza datasets. The novelty and attractiveness of our proposed approach include (i) a new development of Bayesian inference using ANN architectures for ODE based dynamical systems, and (ii) a computationally fast posterior inference by avoiding convergence issues of benchmark Markov Chain Monte Carlo methods. These two features establish the developed approach as an accurate alternative to traditional Bayesian computational methods, with improved computational cost.

Ghost introgression facilitates genomic divergence of a sympatric cryptic lineage in Cycas revoluta.

A cryptic lineage is a genetically diverged but morphologically unrecognized variant of a known species. Clarifying cryptic lineage evolution is essential for quantifying species diversity. In sympatric cryptic lineage divergence compared with allopatric divergence, the forces of divergent selection and mating patterns override geographical isolation. Introgression, by supplying preadapted or neutral standing genetic variations, can promote sympatric cryptic lineage divergence via selection. However, most studies concentrated on extant species introgression, ignoring the genetic legacy of introgression from extinct or unsampled lineages ("ghost introgression"). Cycads are an ideal plant for studying the influence of ghost introgression because of their common interspecific gene flow and past high extinction rate. Here, we utilized reference-based ddRADseq to clarify the role of ghost introgression in the evolution of a previously identified sympatric cryptic lineage in Cycas revoluta. After re-evaluating the evolutionary independency of cryptic lineages, the group-wise diverged single-nucleotide polymorphisms among sympatric and allopatric lineages were compared and functionally annotated. Next, we employed an approximate Bayesian computation method for hypothesis testing to clarify the cryptic lineage evolution and ghost introgression effect. SNPs with the genomic signatures of ghost introgression were further annotated. Our results reconfirmed the evolutionary independency of cryptic lineage among C. revoluta and demonstrated that ghost introgression to the noncryptic lineage facilitated their divergence. Gene function related to heat stress and disease resistance implied ecological adaptation of the main extant populations of C. revoluta.

Integrating Pool-seq uncertainties into demographic inference.

Next-generation sequencing of pooled samples (Pool-seq) is a popular method to assess genome-wide diversity patterns in natural and experimental populations. However, Pool-seq is associated with specific sources of noise, such as unequal individual contributions. Consequently, using Pool-seq for the reconstruction of evolutionary history has remained underexplored. Here we describe a novel Approximate Bayesian Computation (ABC) method to infer demographic history, explicitly modelling Pool-seq sources of error. By jointly modelling Pool-seq data, demographic history and the effects of selection due to barrier loci, we obtain estimates of demographic history parameters accounting for technical errors associated with Pool-seq. Our ABC approach is computationally efficient as it relies on simulating subsets of loci (rather than the whole-genome) and on using relative summary statistics and relative model parameters. Our simulation study results indicate Pool-seq data allows distinction between general scenarios of ecotype formation (single versus parallel origin) and to infer relevant demographic parameters (e.g. effective sizes and split times). We exemplify the application of our method to Pool-seq data from the rocky-shore gastropod Littorina saxatilis, sampled on a narrow geographical scale at two Swedish locations where two ecotypes (Wave and Crab) are found. Our model choice and parameter estimates show that ecotypes formed before colonization of the two locations (i.e. single origin) and are maintained despite gene flow. These results indicate that demographic modelling and inference can be successful based on pool-sequencing using ABC, contributing to the development of suitable null models that allow for a better understanding of the genetic basis of divergent adaptation.

On a class of prior distributions that accounts for uncertainty in the data

A new class of prior distributions that can be used to assess the sensitivity of the Bayesian posterior inference to uncertainty in the data is proposed. This class is derived starting from an initial prior distribution and the likelihood function. We establish the mathematical properties of this class and the conditions under which ordering can be established within this class. We show how the sensitivity analysis can be performed using a standard MCMC procedure for any model whose likelihood, or an approximation, is available in a closed form and illustrate using examples. We also discuss how the proposed class is connected to the main ideas behind the Approximate Bayesian Computation (ABC) method. For this reason, we choose to call the new class of prior distributions as the ABC class of prior distributions. Finally, we close by sketching further possible extensions to this work.