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Approximate Bayesian computation (ABC) is typically used when the likelihood is either unavailable or intractable but where data can be simulated under different parameter settings using a forward model. Despite the recent interest in ABC, high-dimensional data and costly simulations still remain a bottleneck in some applications. There is also no consensus as to how to best assess the performance of such methods without knowing the true posterior. We show how a nonparametric conditional density estimation (CDE) framework, which we refer to as ABC-CDE, help address three nontrivial challenges in ABC: (i) how to efficiently estimate the posterior distribution with limited simulations and different types of data, (ii) how to tune and compare the performance of ABC and related methods in estimating the posterior itself, rather than just certain properties of the density, and (iii) how to efficiently choose among a large set of summary statistics based on a CDE surrogate loss. We provide theoretical and empirical evidence that justify ABC-CDE procedures that directly estimate and assess the posterior based on an initial ABC sample, and we describe settings where standard ABC and regression-based approaches are inadequate. Supplemental materials for this article are available online.

Deciphering invasion routes from molecular data is crucial to understanding biological invasions, including identifying bottlenecks in population size and admixture among distinct populations. Here, we unravel the invasion routes of the invasive pest Drosophila suzukii using a multi-locus microsatellite dataset (25 loci on 23 worldwide sampling locations). To do this, we use approximate Bayesian computation (ABC), which has improved the reconstruction of invasion routes, but can be computationally expensive. We use our study to illustrate the use of a new, more efficient, ABC method, ABC random forest (ABC-RF) and compare it to a standard ABC method (ABC-LDA). We find that Japan emerges as the most probable source of the earliest recorded invasion into Hawaii. Southeast China and Hawaii together are the most probable sources of populations in western North America, which then in turn served as sources for those in eastern North America. European populations are genetically more homogeneous than North American populations, and their most probable source is northeast China, with evidence of limited gene flow from the eastern US as well. All introduced populations passed through bottlenecks, and analyses reveal five distinct admixture events. These findings can inform hypotheses concerning how this species evolved between different and independent source and invasive populations. Methodological comparisons indicate that ABC-RF and ABC-LDA show concordant results if ABC-LDA is based on a large number of simulated datasets but that ABC-RF out-performs ABC-LDA when using a comparable and more manageable number of simulated datasets, especially when analyzing complex introduction scenarios.

Linking pattern to process across spatial and temporal scales has been a key goal of the field of biogeography. In January 2017, the 8th biennial conference of the International Biogeography Society sponsored a symposium on Building up biogeography—process to pattern that aimed to review progress towards this goal. Here we present a summary of the symposium, in which we identified promising areas of current research and suggested future research directions. We focus on (1) emerging types of data such as behavioural observations and ancient DNA, (2) how to better incorporate historical data (such as fossils) to move beyond what we term \"footprint measures\" of past dynamics and (3) the role that novel modelling approaches (e.g. maximum entropy theory of ecology and approximate Bayesian computation) and conceptual frameworks can play in the unification of disciplines. We suggest that the gaps separating pattern and process are shrinking, and that we can better bridge these aspects by considering the dimensions of space and time simultaneously.

A growing number of generative statistical models do not permit the numerical evaluation of their likelihood functions. Approximate Bayesian computation has become a popular approach to overcome this issue, in which one simulates synthetic data sets given parameters and compares summaries of these data sets with the corresponding observed values. We propose to avoid the use of summaries and the ensuing loss of information by instead using the Wasserstein distance between the empirical distributions of the observed and synthetic data. This generalizes the well-known approach of using order statistics within approximate Bayesian computation to arbitrary dimensions. We describe how recently developed approximations of the Wasserstein distance allow the method to scale to realistic data sizes, and we propose a new distance based on the Hilbert space filling curve. We provide a theoretical study of the method proposed, describing consistency as the threshold goes to 0 while the observations are kept fixed, and concentration properties as the number of observations grows. Various extensions to time series data are discussed. The approach is illustrated on various examples, including univariate and multivariate g-and-k distributions, a toggle switch model from systems biology, a queuing model and a Lévy-driven stochastic volatility model.

Having the ability to work with complex models can be highly beneficial. However, complex models often have intractable likelihoods, so methods that involve evaluation of the likelihood function are infeasible. In these situations, the benefits of working with likelihood-free methods become apparent. Likelihood-free methods, such as parametric Bayesian indirect likelihood that uses the likelihood of an alternative parametric auxiliary model, have been explored throughout the literature as a viable alternative when the model of interest is complex. One of these methods is called the synthetic likelihood (SL), which uses a multivariate normal approximation of the distribution of a set of summary statistics. This article explores the accuracy and computational efficiency of the Bayesian version of the synthetic likelihood (BSL) approach in comparison to a competitor known as approximate Bayesian computation (ABC) and its sensitivity to its tuning parameters and assumptions. We relate BSL to pseudo-marginal methods and propose to use an alternative SL that uses an unbiased estimator of the SL, when the summary statistics have a multivariate normal distribution. Several applications of varying complexity are considered to illustrate the findings of this article.

Many domains of science have developed complex simulations to describe phenomena of interest. While these simulations provide high-fidelity models, they are poorly suited for inference and lead to challenging inverse problems. We review the rapidly developing field of simulation-based inference and identify the forces giving additional momentum to the field. Finally, we describe how the frontier is expanding so that a broad audience can appreciate the profound influence these developments may have on science.

Abstract Bayesian inference plays an important role in phylogenetics, evolutionary biology, and in many other branches of science. It provides a principled framework for dealing with uncertainty and quantifying how it changes in the light of new evidence. For many complex models and inference problems, however, only approximate quantitative answers are obtainable. Approximate Bayesian computation (ABC) refers to a family of algorithms for approximate inference that makes a minimal set of assumptions by only requiring that sampling from a model is possible. We explain here the fundamentals of ABC, review the classical algorithms, and highlight recent developments. [ABC; approximate Bayesian computation; Bayesian inference; likelihood-free inference; phylogenetics; simulator-based models; stochastic simulation models; tree-based models.]

Persian walnut (Juglans regia) is cultivated worldwide for its high-quality wood and nuts, but its origin has remained mysterious because in phylogenies it occupies an unresolved position between American black walnuts and Asian butternuts. Equally unclear is the origin of the only American butternut, J. cinerea. We resequenced the whole genome of 80 individuals from 19 of the 22 species of Juglans and assembled the genome of its relatives Pterocarya stenoptera and Platycarya strobilacea. Using phylogenetic-network analysis of single-copy nuclear genes, genome-wide site pattern probabilities, and Approximate Bayesian Computation, we discovered that J. regia (and its landrace J. sigillata) arose as a hybrid between the American and the Asian lineages and that J. cinerea resulted from massive introgression from an immigrating Asian butternut into the genome of an American black walnut. Approximate Bayesian Computation modeling placed the hybrid origin in the late Pliocene, ∼3.45 My, with both parental lineages since having gone extinct in Europe.

This paper introduces a new way to compact a continuous probability distribution F into a set of representative points called support points. These points are obtained by minimizing the energy distance, a statistical potential measure initially proposed by Székely and Rizzo [InterStat 5 (2004) 1–6] for testing goodness-of-fit. The energy distance has two appealing features. First, its distance-based structure allows us to exploit the duality between powers of the Euclidean distance and its Fourier transform for theoretical analysis. Using this duality, we show that support points converge in distribution to F, and enjoy an improved error rate to Monte Carlo for integrating a large class of functions. Second, the minimization of the energy distance can be formulated as a difference-of-convex program, which we manipulate using two algorithms to efficiently generate representative point sets. In simulation studies, support points provide improved integration performance to both Monte Carlo and a specific quasi-Monte Carlo method. Two important applications of support points are then highlighted: (a) as a way to quantify the propagation of uncertainty in expensive simulations and (b) as a method to optimally compact Markov chain Monte Carlo (MCMC) samples in Bayesian computation.

We consider the computational and statistical issues for high-dimensional Bayesian model selection under the Gaussian spike and slab priors. To avoid large matrix computations needed in a standard Gibbs sampler, we propose a novel Gibbs sampler called \"Skinny Gibbs\" which is much more scalable to high-dimensional problems, both in memory and in computational efficiency. In particular, its computational complexity grows only linearly in p, the number of predictors, while retaining the property of strong model selection consistency even when p is much greater than the sample size n. The present article focuses on logistic regression due to its broad applicability as a representative member of the generalized linear models. We compare our proposed method with several leading variable selection methods through a simulation study to show that Skinny Gibbs has a strong performance as indicated by our theoretical work. Supplementary materials for this article are available online.