We propose a novel approach to approximate Bayesian computation (ABC) that seeks to cater for possible misspecification of the assumed model. This new approach can be equally applied to rejection-based ABC and to popular regression adjustment ABC. We demonstrate that this new approach mitigates the poor performance of regression adjusted ABC that can eventuate when the model is misspecified. In addition, this new adjustment approach allows us to detect which features of the observed data can not be reliably reproduced by the assumed model. A series of simulated and empirical examples illustrate this new approach.
Bayes linear analysis and approximate Bayesian computation (ABC) are techniques commonly used in the Bayesian analysis of complex models. In this article we connect these ideas by demonstrating that regression-adjustment ABC algorithms produce samples for which first and second order moment summaries approximate adjusted expectation and variance for a Bayes linear analysis. This gives regression-adjustment methods a useful interpretation and role in exploratory analysis in high-dimensional problems. As a result, we propose a new method for combining high-dimensional, regression-adjustment ABC …
This report is a collection of comments on the Read Paper of Fearnhead and Prangle (2011), to appear in the Journal of the Royal Statistical Society Series B, along with a reply from the authors.
The situation frequently arises where working with the likelihood function is problematic. This can happen for several reasons---perhaps the likelihood is prohibitively computationally expensive, perhaps it lacks some robustness property, or perhaps it is simply not known for the model under consideration. In these cases, it is often possible to specify alternative functions of the parameters and the data that can be maximized to obtain asymptotically normal estimates. However, these scenarios present obvious problems if one is interested in applying Bayesian techniques. Here we describe open-…
A new recalibration post-processing method is presented to improve the quality of the posterior approximation when using Approximate Bayesian Computation (ABC) algorithms. Recalibration may be used in conjunction with existing post-processing methods, such as regression-adjustments. In addition, this work extends and strengthens the links between ABC and indirect inference algorithms, allowing more extensive use of misspecified auxiliary models in the ABC context. The method is illustrated using simulated examples to demonstrate the effects of recalibration under various conditions, and throug…
By the nature of their construction, many statistical models for extremes result in likelihood functions that are computationally prohibitive to evaluate. This is consequently problematic for the purposes of likelihood-based inference. With a focus on the Bayesian framework, this chapter examines the use of approximate Bayesian computation (ABC) techniques for the fitting and analysis of statistical models for extremes. After introducing the ideas behind ABC algorithms and methods, we demonstrate their application to extremal models in stereology and spatial extremes.
This Chapter, "Overview of Approximate Bayesian Computation", is to appear as the first chapter in the forthcoming Handbook of Approximate Bayesian Computation (2018). It details the main ideas and concepts behind ABC methods with many examples and illustrations.
Approximate Bayesian computation methods can be used to evaluate posterior distributions without having to calculate likelihoods. In this paper we discuss and apply an approximate Bayesian computation (ABC) method based on sequential Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC SMC gives 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 infe…
This chapter will appear in the forthcoming Handbook of Approximate Bayesian Computation (2018). The conceptual and methodological framework that underpins approximate Bayesian computation (ABC) is targetted primarily towards problems in which the likelihood is either challenging or missing. ABC uses a simulation-based non-parametric estimate of the likelihood of a summary statistic and assumes that the generation of data from the model is computationally cheap. This chapter reviews two alternative approaches for estimating the intractable likelihood, with the goal of reducing the necessary mo…
This paper introduces a framework for speeding up Bayesian inference conducted in presence of large datasets. We design a Markov chain whose transition kernel uses an (unknown) fraction of (fixed size) of the available data that is randomly refreshed throughout the algorithm. Inspired by the Approximate Bayesian Computation (ABC) literature, the subsampling process is guided by the fidelity to the observed data, as measured by summary statistics. The resulting algorithm, Informed Sub-Sampling MCMC (ISS-MCMC), is a generic and flexible approach which, contrary to existing scalable methodologies…
Simulation models often lack tractable likelihood functions, making likelihood-free inference methods indispensable. Approximate Bayesian computation generates likelihood-free posterior samples by comparing simulated and observed data through some distance measure, but existing approaches are often poorly suited to time series simulators, for example due to an independent and identically distributed data assumption. In this paper, we propose to use path signatures in approximate Bayesian computation to handle the sequential nature of time series. We provide theoretical guarantees on the result…
Likelihood-free methods, such as approximate Bayesian computation, are powerful tools for practical inference problems with intractable likelihood functions. Markov chain Monte Carlo and sequential Monte Carlo variants of approximate Bayesian computation can be effective techniques for sampling posterior distributions in an approximate Bayesian computation setting. However, without careful consideration of convergence criteria and selection of proposal kernels, such methods can lead to very biased inference or computationally inefficient sampling. In contrast, rejection sampling for approximat…
In this paper, the parameter estimation of ARMA(p,q) model is given by approximate Bayesian computation algorithm. In order to improve the sampling efficiency of the algorithm, approximate Bayesian computation should select as many statistics as possible with parameter information in low dimension. Firstly, we use the autocorrelation coefficient of the first p+q order sample as the statistic and obtain an approximate Bayesian estimation of the AR coefficient, transforming the ARMA(p,q) model into the MA(q) model. Considering the first q order sample autocorrelation functions and sample varianc…
This Chapter, "A Guide to General-Purpose ABC Software", is to appear in the forthcoming Handbook of Approximate Bayesian Computation (2018). We present general-purpose software to perform Approximate Bayesian Computation (ABC) as implemented in the R-packages abc and EasyABC and the c++ program ABCtoolbox. With simple toy models we demonstrate how to perform parameter inference, model selection, validation and optimal choice of summary statistics. We demonstrate how to combine ABC with Markov Chain Monte Carlo and describe a realistic population genetics application.
With larger data at their disposal, scientists are emboldened to tackle complex questions that require sophisticated statistical models. It is not unusual for the latter to have likelihood functions that elude analytical formulations. Even under such adversity, when one can simulate from the sampling distribution, Bayesian analysis can be conducted using approximate methods such as Approximate Bayesian Computation (ABC) or Bayesian Synthetic Likelihood (BSL). A significant drawback of these methods is that the number of required simulations can be prohibitively large, thus severely limiting th…
ABC (approximate Bayesian computation) is a general approach for dealing with models with an intractable likelihood. In this work, we derive ABC algorithms based on QMC (quasi- Monte Carlo) sequences. We show that the resulting ABC estimates have a lower variance than their Monte Carlo counter-parts. We also develop QMC variants of sequential ABC algorithms, which progressively adapt the proposal distribution and the acceptance threshold. We illustrate our QMC approach through several examples taken from the ABC literature. Keywords: Approximate Bayesian computation, Likelihood-free inference,…
We present a procedure to diagnose model misspecification in situations where inference is performed using approximate Bayesian computation. We demonstrate theoretically, and empirically that this procedure can consistently detect the presence of model misspecification. Our examples demonstrates that this approach delivers good finite-sample performance and is computational less onerous than existing approaches, all of which require re-running the inference algorithm. An empirical application to modelling exchange rate log returns using a g-and-k distribution completes the paper.
Sequential techniques can enhance the efficiency of the approximate Bayesian computation algorithm, as in Sisson et al.'s (2007) partial rejection control version. While this method is based upon the theoretical works of Del Moral et al. (2006), the application to approximate Bayesian computation results in a bias in the approximation to the posterior. An alternative version based on genuine importance sampling arguments bypasses this difficulty, in connection with the population Monte Carlo method of Cappe et al. (2004), and it includes an automatic scaling of the forward kernel. When applied…
Approximate Bayesian computation allows for inference of complicated probabilistic models with intractable likelihoods using model simulations. The Markov chain Monte Carlo implementation of approximate Bayesian computation is often sensitive to the tolerance parameter: low tolerance leads to poor mixing and large tolerance entails excess bias. We consider an approach using a relatively large tolerance for the Markov chain Monte Carlo sampler to ensure its sufficient mixing, and post-processing the output leading to estimators for a range of finer tolerances. We introduce an approximate confid…
In the following article we consider approximate Bayesian computation (ABC) inference. We introduce a method for numerically approximating ABC posteriors using the multilevel Monte Carlo (MLMC). A sequential Monte Carlo version of the approach is developed and it is shown under some assumptions that for a given level of mean square error, this method for ABC has a lower cost than i.i.d. sampling from the most accurate ABC approximation. Several numerical examples are given.
Approximate Bayesian Computation (ABC) is a statistical learning technique to calibrate and select models by comparing observed data to simulated data. This technique bypasses the use of the likelihood and requires only the ability to generate synthetic data from the models of interest. We apply ABC to fit and compare insurance loss models using aggregated data. A state-of-the-art ABC implementation in Python is proposed. It uses sequential Monte Carlo to sample from the posterior distribution and the Wasserstein distance to compare the observed and synthetic data.
Stochastic gradient Hamiltonian Monte Carlo (SGHMC) is an efficient method for sampling from continuous distributions. It is a faster alternative to HMC: instead of using the whole dataset at each iteration, SGHMC uses only a subsample. This improves performance, but introduces bias that can cause SGHMC to converge to the wrong distribution. One can prevent this using a step size that decays to zero, but such a step size schedule can drastically slow down convergence. To address this tension, we propose a novel second-order SG-MCMC algorithm---AMAGOLD---that infrequently uses Metropolis-Hastin…
Multifidelity approximate Bayesian computation (MF-ABC) is a likelihood-free technique for parameter inference that exploits model approximations to significantly increase the speed of ABC algorithms (Prescott and Baker, 2020). Previous work has considered MF-ABC only in the context of rejection sampling, which does not explore parameter space particularly efficiently. In this work, we integrate the multifidelity approach with the ABC sequential Monte Carlo (ABC-SMC) algorithm into a new MF-ABC-SMC algorithm. We show that the improvements generated by each of ABC-SMC and MF-ABC to the efficien…
Approximate Bayesian computation (ABC) has gained popularity in recent years owing to its easy implementation, nice interpretation and good performance. Its advantages are more visible when one encounters complex models where maximum likelihood estimation as well as Bayesian analysis via Markov chain Monte Carlo demand prohibitively large amount of time. This paper examines properties of ABC both from a theoretical as well as from a computational point of view.We consolidate the ABC theory by proving theorems related to its limiting behaviour. In particular, we consider partial posteriors, whi…
In the following article we consider approximate Bayesian computation (ABC) for certain classes of time series models. In particular, we focus upon scenarios where the likelihoods of the observations and parameter are intractable, by which we mean that one cannot evaluate the likelihood even up-to a positive unbiased estimate. This paper reviews and develops a class of approximation procedures based upon the idea of ABC, but, specifically maintains the probabilistic structure of the original statistical model. This idea is useful, in that it can facilitate an analysis of the bias of the approx…
Light and Widely Applicable (LWA-) MCMC is a novel approximation of the Metropolis-Hastings kernel targeting a posterior distribution defined on a large number of observations. Inspired by Approximate Bayesian Computation, we design a Markov chain whose transition makes use of an unknown but fixed, fraction of the available data, where the random choice of sub-sample is guided by the fidelity of this sub-sample to the observed data, as measured by summary (or sufficient) statistics. LWA-MCMC is a generic and flexible approach, as illustrated by the diverse set of examples which we explore. In …
We present asymptotic results for the regression-adjusted version of approximate Bayesian computation introduced by Beaumont(2002). We show that for an appropriate choice of the bandwidth, regression adjustment will lead to a posterior that, asymptotically, correctly quantifies uncertainty. Furthermore, for such a choice of bandwidth we can implement an importance sampling algorithm to sample from the posterior whose acceptance probability tends to unity as the data sample size increases. This compares favourably to results for standard approximate Bayesian computation, where the only way to o…
This document is due to appear as a chapter of the forthcoming Handbook of Approximate Bayesian Computation (ABC) edited by S. Sisson, Y. Fan, and M. Beaumont. Since the earliest work on ABC, it has been recognised that using summary statistics is essential to produce useful inference results. This is because ABC suffers from a curse of dimensionality effect, whereby using high dimensional inputs causes large approximation errors in the output. It is therefore crucial to find low dimensional summaries which are informative about the parameter inference or model choice task at hand. This chapte…
Controlled branching processes are stochastic growth population models in which the number of individuals with reproductive capacity in each generation is controlled by a random control function. The purpose of this work is to examine the Approximate Bayesian Computation (ABC) methods and to propose appropriate summary statistics for them in the context of these processes. This methodology enables to approximate the posterior distribution of the parameters of interest satisfactorily without explicit likelihood calculations and under a minimal set of assumptions. In particular, the tolerance re…
A two-stage approach is proposed to overcome the problem in quantile regression, where separately fitted curves for several quantiles may cross. The standard Bayesian quantile regression model is applied in the first stage, followed by a Gaussian process regression adjustment, which monotonizes the quantile function whilst borrowing strength from nearby quantiles. The two stage approach is computationally efficient, and more general than existing techniques. The method is shown to be competitive with alternative approaches via its performance in simulated examples.
Approximate Bayesian inference on the basis of summary statistics is well-suited to complex problems for which the likelihood is either mathematically or computationally intractable. However the methods that use rejection suffer from the curse of dimensionality when the number of summary statistics is increased. Here we propose a machine-learning approach to the estimation of the posterior density by introducing two innovations. The new method fits a nonlinear conditional heteroscedastic regression of the parameter on the summary statistics, and then adaptively improves estimation using import…
Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian inference in such cases. We present results on the asymptotic variance of estimators obtained using approximate Bayesian computation in a large-data limit. Our key assumption is that the data are summarized by a fixed-dimensional summary statistic that obeys a central limit theorem. We prove asymptotic normality of the mean of the approximate Bayesian computation …
We present a sequential Monte Carlo sampler variant of the partial rejection control algorithm, and show that this variant can be considered as a sequential Monte Carlo sampler with a modified mutation kernel. We prove that the new sampler can reduce the variance of the incremental importance weights when compared with standard sequential Monte Carlo samplers. We provide a study of theoretical properties of the new algorithm, and make connections with some existing algorithms. Finally, the sampler is adapted for application under the challenging "likelihood free," approximate Bayesian computat…
We analyze the computational efficiency of approximate Bayesian computation (ABC), which approximates a likelihood function by drawing pseudo-samples from the associated model. For the rejection sampling version of ABC, it is known that multiple pseudo-samples cannot substantially increase (and can substantially decrease) the efficiency of the algorithm as compared to employing a high-variance estimate based on a single pseudo-sample. We show that this conclusion also holds for a Markov chain Monte Carlo version of ABC, implying that it is unnecessary to tune the number of pseudo-samples used …
We derive the optimal proposal density for Approximate Bayesian Computation (ABC) using Sequential Monte Carlo (SMC) (or Population Monte Carlo, PMC). The criterion for optimality is that the SMC/PMC-ABC sampler maximise the effective number of samples per parameter proposal. The optimal proposal density represents the optimal trade-off between favoring high acceptance rate and reducing the variance of the importance weights of accepted samples. We discuss two convenient approximations of this proposal and show that the optimal proposal density gives a significant boost in the expected samplin…
In many instances, the application of approximate Bayesian methods is hampered by two practical features: 1) the requirement to project the data down to low-dimensional summary, including the choice of this projection, which ultimately yields inefficient inference; 2) a possible lack of robustness to deviations from the underlying model structure. Motivated by these efficiency and robustness concerns, we construct a new Bayesian method that can deliver efficient estimators when the underlying model is well-specified, and which is simultaneously robust to certain forms of model misspecification…
This book chapter introduces regression approaches and regression adjustment for Approximate Bayesian Computation (ABC). Regression adjustment adjusts parameter values after rejection sampling in order to account for the imperfect match between simulations and observations. Imperfect match between simulations and observations can be more pronounced when there are many summary statistics, a phenomenon coined as the curse of dimensionality. Because of this imperfect match, credibility intervals obtained with regression approaches can be inflated compared to true credibility intervals. The chapte…
Approximate Bayesian computation (ABC) is a widely used inference method in Bayesian statistics to bypass the point-wise computation of the likelihood. In this paper we develop theoretical bounds for the distance between the statistics used in ABC. We show that some versions of ABC are inherently robust to misspecification. The bounds are given in the form of oracle inequalities for a finite sample size. The dependence on the dimension of the parameter space and the number of statistics is made explicit. The results are shown to be amenable to oracle inequalities in parameter space. We apply o…
We combine Bayesian prediction and weighted inference as a unified approach to survey inference. The general principles of Bayesian analysis imply that models for survey outcomes should be conditional on all variables that affect the probability of inclusion. We incorporate the weighting variables under the framework of multilevel regression and poststratification, as a byproduct generating model-based weights after smoothing. We investigate deep interactions and introduce structured prior distributions for smoothing and stability of estimates. The computation is done via Stan and implemented …
We propose a new methodology to perform mineralogic inversion from wellbore logs based on a Bayesian linear regression model. Our method essentially relies on three steps. The first step makes use of Approximate Bayesian Computation (ABC) and selects from the Bayesian generator a set of candidates-volumes corresponding closely to the wellbore data responses. The second step gathers these candidates through a density-based clustering algorithm. A mineral scenario is assigned to each cluster through direct mineralogical inversion, and we provide a confidence estimate for each lithological hypoth…
How can heuristic strategies emerge from smaller building blocks? We propose Approximate Bayesian Computation as a computational solution to this problem. As a first proof of concept, we demonstrate how a heuristic decision strategy such as Take The Best (TTB) can be learned from smaller, probabilistically updated building blocks. Based on a self-reinforcing sampling scheme, different building blocks are combined and, over time, tree-like non-compensatory heuristics emerge. This new algorithm, coined Approximately Bayesian Computed Take The Best (ABC-TTB), is able to recover a data set that wa…
Approximate Bayesian Computation (ABC for short) is a family of computational techniques which offer an almost automated solution in situations where evaluation of the posterior likelihood is computationally prohibitive, or whenever suitable likelihoods are not available. In the present paper, we analyze the procedure from the point of view of k-nearest neighbor theory and explore the statistical properties of its outputs. We discuss in particular some asymptotic features of the genuine conditional density estimate associated with ABC, which is an interesting hybrid between a k-nearest neighbo…
We study Bayesian approaches to causal inference via propensity score regression. Much of the Bayesian literature on propensity score methods have relied on approaches that cannot be viewed as fully Bayesian in the context of conventional `likelihood times prior' posterior inference; in addition, most methods rely on parametric and distributional assumptions, and presumed correct specification. We emphasize that causal inference is typically carried out in settings of mis-specification, and develop strategies for fully Bayesian inference that reflect this. We focus on methods based on decision…
This paper proposes a new Bayesian approach to estimate the Gini coefficient from the Lorenz curve based on grouped data. The proposed approach assumes a hypothetical income distribution and estimates the parameter by directly working on the likelihood function implied by the Lorenz curve of the income distribution from the grouped data. It inherits the advantages of two existing approaches through which the Gini coefficient can be estimated more accurately and a straightforward interpretation about the underlying income distribution is provided. Since the likelihood function is implicitly def…
Historical data about disease outcomes can be integrated into the analysis of clinical trials in many ways. We build on existing literature that uses prognostic scores from a predictive model to increase the efficiency of treatment effect estimates via covariate adjustment. Here we go further, utilizing a Bayesian framework that combines prognostic covariate adjustment with an empirical prior distribution learned from the predictive performances of the prognostic model on past trials. The Bayesian approach interpolates between prognostic covariate adjustment with strict type I error control wh…
A new method is proposed for the correction of confidence intervals when the original interval does not have the correct nominal coverage probabilities in the frequentist sense. The proposed method is general and does not require any distributional assumptions. It can be applied to both frequentist and Bayesian inference where interval estimates are desired. We provide theoretical results for the consistency of the proposed estimator, and give two complex examples, on confidence interval correction for composite likelihood estimators and in approximate Bayesian computation (ABC), to demonstrat…
Approximate Bayesian computation (ABC) using a sequential Monte Carlo method provides a comprehensive platform for parameter estimation, model selection and sensitivity analysis in differential equations. However, this method, like other Monte Carlo methods, incurs a significant computational cost as it requires explicit numerical integration of differential equations to carry out inference. In this paper we propose a novel method for circumventing the requirement of explicit integration by using derivatives of Gaussian processes to smooth the observations from which parameters are estimated. …
We study batch normalisation in the context of variational inference methods in Bayesian neural networks, such as mean-field or MC Dropout. We show that batch-normalisation does not affect the optimum of the evidence lower bound (ELBO). Furthermore, we study the Monte Carlo Batch Normalisation (MCBN) algorithm, proposed as an approximate inference technique parallel to MC Dropout, and show that for larger batch sizes, MCBN fails to capture epistemic uncertainty. Finally, we provide insights into what is required to fix this failure, namely having to view the mini-batch size as a variational pa…
Analysis of observational studies increasingly confronts the challenge of determining which of a possibly high-dimensional set of available covariates are required to satisfy the assumption of ignorable treatment assignment for estimation of causal effects. We propose a Bayesian nonparametric approach that simultaneously 1) prioritizes inclusion of adjustment variables in accordance with existing principles of confounder selection; 2) estimates causal effects in a manner that permits complex relationships among confounders, exposures, and outcomes; and 3) provides causal estimates that account…
Until recently, the use of Bayesian inference in population genetics was limited to a few cases because for many realistic population genetic models the likelihood function cannot be calculated analytically . The situation changed with the advent of likelihood-free inference algorithms, often subsumed under the term Approximate Bayesian Computation (ABC). A key innovation was the use of a post-sampling regression adjustment, allowing larger tolerance values and as such shifting computation time to realistic orders of magnitude (see Beaumont et al., 2002). Here we propose a reformulation of the…
A key challenge in estimating the infection fatality rate (IFR) -- and its relation with various factors of interest -- is determining the total number of cases. The total number of cases is not known because not everyone is tested, but also, more importantly, because tested individuals are not representative of the population at large. We refer to the phenomenon whereby infected individuals are more likely to be tested than non-infected individuals, as "preferential testing." An open question is whether or not it is possible to reliably estimate the IFR without any specific knowledge about th…
Randomized experiments play a major role in data-driven decision making across many different fields and disciplines. In medicine, for example, randomized controlled trials (RCTs) are the backbone of clinical trial methodology for testing the efficacy of new drugs and therapies versus existing treatments or placebo. In business and marketing, randomized experiments are typically referred to as A/B tests when there are only two arms, or variants, in the experiment, and as multivariate A/B tests when there are more than two arms. Typical applications of A/B tests include comparing the effectiven…
Identification of model parameters in computer simulations is an important topic in computer experiments. We propose a new method, called the projected kernel calibration method, to estimate these model parameters. The proposed method is proven to be asymptotic normal and semi-parametric efficient. As a frequentist method, the proposed method is as efficient as the $L_2$ calibration method proposed by Tuo and Wu [Ann. Statist. 43 (2015) 2331-2352]. On the other hand, the proposed method has a natural Bayesian version, which the $L_2$ method does not have. This Bayesian version allows users to …
