A new approach to choose acceptance cutoff for approximate Bayesian computation

The approximate Bayesian computation (ABC) algorithm is used to estimate parameters from complicated phenomena, where likelihood is intractable. Here, we report the development of an algorithm to choose the tolerance level for ABC. We have illustrated the performance of our proposed method by simulating the estimation of scaled mutation and recombination rates. The result shows that the proposed algorithm performs well.     

Filtering via approximate Bayesian computation

Approximate Bayesian computation (ABC) has become a popular technique to facilitate Bayesian inference from complex models. In this article we present an ABC approximation designed to perform biased filtering for a Hidden Markov Model when the likelihood function is intractable. We use a sequential Monte Carlo (SMC) algorithm to both fit and sample from our ABC approximation of the target probability density. This approach is shown to, empirically, be more accurate w.r.t.?the original filter than competing methods. The theoretical bias of our method is investigated; it is shown that the bias goes to zero at the expense of increased computational effort. Our approach is illustrated on a constrained sequential lasso for portfolio allocation to 15 constituents of the FTSE 100 share index.     

Long-range dependence and approximate Bayesian computation

In this work, we propose a method for estimating the Hurst index, or memory parameter, of a stationary process with long memory in a Bayesian fashion. Such approach provides an approximation for the posterior distribution for the memory parameter and it is based on a simple application of the so-called approximate Bayesian computation (ABC), also known as likelihood-free method. Some popular existing estimators are reviewed and compared to this method for the fractional Brownian motion, for a long-range binary process and for the Rosenblatt process. The performance of our proposal is remarkably efficient.     

Generalized multiple-point Metropolis algorithms for approximate Bayesian computation

It is well known that the approximate Bayesian computation algorithm based on Markov chain Monte Carlo methods suffers from the sensitivity to the choice of starting values, inefficiency and a low acceptance rate. To overcome these problems, this study proposes a generalization of the multiple-point Metropolis algorithm, which proceeds by generating multiple-dependent proposals and then by selecting a candidate among the set of proposals on the basis of weights that can be chosen arbitrarily. The performance of the proposed algorithm is illustrated by using both simulated and real data.     

Reverse engineering gene regulatory networks using approximate Bayesian computation

Gene regulatory networks are collections of genes that interact with one other and with other substances in the cell. By measuring gene expression over time using high-throughput technologies, it may be possible to reverse engineer, or infer, the structure of the gene network involved in a particular cellular process. These gene expression data typically have a high dimensionality and a limited number of biological replicates and time points. Due to these issues and the complexity of biological systems, the problem of reverse engineering networks from gene expression data demands a specialized suite of statistical tools and methodologies. We propose a non-standard adaptation of a simulation-based approach known as Approximate Bayesian Computing based on Markov chain Monte Carlo sampling. This approach is particularly well suited for the inference of gene regulatory networks from longitudinal data. The performance of this approach is investigated via simulations and using longitudinal expression data from a genetic repair system in Escherichia coli.     

The use of a single pseudo-sample in approximate Bayesian computation

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 in ABC-MCMC. This conclusion is in contrast to particle MCMC methods, for which increasing the number of particles can provide large gains in computational efficiency.     

Diagnostic tools for approximate Bayesian computation using the coverage property

Approximate Bayesian computation (ABC) is an approach to sampling from an approximate posterior distribution in the presence of a computationally intractable likelihood function. A common implementation is based on simulating model, parameter and dataset triples from the prior, and then accepting as samples from the approximate posterior, those model and parameter pairs for which the corresponding dataset, or a summary of that dataset, is ‘close’ to the observed data. Closeness is typically determined though a distance measure and a kernel scale parameter. Appropriate choice of that parameter is important in producing a good quality approximation. This paper proposes diagnostic tools for the choice of the kernel scale parameter based on assessing the coverage property, which asserts that credible intervals have the correct coverage levels in appropriately designed simulation settings. We provide theoretical results on coverage for both model and parameter inference, and adapt these into diagnostics for the ABC context. We re‐analyse a study on human demographic history to determine whether the adopted posterior approximation was appropriate. Code implementing the proposed methodology is freely available in the R package abctools .     

Fast approximate Bayesian computation for estimating parameters in differential equations

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 evaluate our methods using synthetic data generated from model biological systems described by ordinary and delay differential equations. Upon comparing the performance of our method to existing ABC techniques, we demonstrate that it produces comparably reliable parameter estimates at a significantly reduced execution time.     

An adaptive sequential Monte Carlo method for approximate Bayesian computation

Approximate Bayesian computation (ABC) is a popular approach to address inference problems where the likelihood function is intractable, or expensive to calculate. To improve over Markov chain Monte Carlo (MCMC) implementations of ABC, the use of sequential Monte Carlo (SMC) methods has recently been suggested. Most effective SMC algorithms that are currently available for ABC have a computational complexity that is quadratic in the number of Monte Carlo samples (Beaumont et al., Biometrika 86:983?C990, 2009; Peters et al., Technical report, 2008; Toni et al., J.?Roy. Soc. Interface 6:187?C202, 2009) and require the careful choice of simulation parameters. In this article an adaptive SMC algorithm is proposed which admits a computational complexity that is linear in the number of samples and adaptively determines the simulation parameters. We demonstrate our algorithm on a toy example and on a birth-death-mutation model arising in epidemiology.     

A Bayesian approach with generalized ridge estimation for high-dimensional regression and testing

This paper adopts a Bayesian strategy for generalized ridge estimation for high-dimensional regression. We also consider significance testing based on the proposed estimator, which is useful for selecting regressors. Both theoretical and simulation studies show that the proposed estimator can simultaneously outperform the ordinary ridge estimator and the LSE in terms of the mean square error (MSE) criterion. The simulation study also demonstrates the competitive MSE performance of our proposal with the Lasso under sparse models. We demonstrate the method using the lung cancer data involving high-dimensional microarrays.     

Choosing summary statistics by least angle regression for approximate Bayesian computation

Bayesian statistical inference relies on the posterior distribution. Depending on the model, the posterior can be more or less difficult to derive. In recent years, there has been a lot of interest in complex settings where the likelihood is analytically intractable. In such situations, approximate Bayesian computation (ABC) provides an attractive way of carrying out Bayesian inference. For obtaining reliable posterior estimates however, it is important to keep the approximation errors small in ABC. The choice of an appropriate set of summary statistics plays a crucial role in this effort. Here, we report the development of a new algorithm that is based on least angle regression for choosing summary statistics. In two population genetic examples, the performance of the new algorithm is better than a previously proposed approach that uses partial least squares.     

On sequential Monte Carlo, partial rejection control and approximate Bayesian computation

We present a variant of the sequential Monte Carlo sampler by incorporating the partial rejection control mechanism of Liu (2001). We show that the resulting algorithm 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, and provide a central limit theorem. Finally, the sampler is adapted for application under the challenging approximate Bayesian computation modelling framework.     

Continuous event monitoring via a Bayesian predictive approach

In clinical trials, continuous monitoring of event incidence rate plays a critical role in making timely decisions affecting trial outcome. For example, continuous monitoring of adverse events protects the safety of trial participants, while continuous monitoring of efficacy events helps identify early signals of efficacy or futility. Because the endpoint of interest is often the event incidence associated with a given length of treatment duration (e.g., incidence proportion of an adverse event with 2 years of dosing), assessing the event proportion before reaching the intended treatment duration becomes challenging, especially when the event onset profile evolves over time with accumulated exposure. In particular, in the earlier part of the study, ignoring censored subjects may result in significant bias in estimating the cumulative event incidence rate. Such a problem is addressed using a predictive approach in the Bayesian framework. In the proposed approach, experts' prior knowledge about both the frequency and timing of the event occurrence is combined with observed data. More specifically, during any interim look, each event‐free subject will be counted with a probability that is derived using prior knowledge. The proposed approach is particularly useful in early stage studies for signal detection based on limited information. But it can also be used as a tool for safety monitoring (e.g., data monitoring committee) during later stage trials. Application of the approach is illustrated using a case study where the incidence rate of an adverse event is continuously monitored during an Alzheimer's disease clinical trial. The performance of the proposed approach is also assessed and compared with other Bayesian and frequentist methods via simulation. Copyright © 2015 John Wiley & Sons, Ltd.     

A semiparametric Bayesian approach for joint modeling of longitudinal trait and event time

Inference on the whole biological system is the recent focus in bioscience. Different biomarkers, although seem to function separately, can actually control some event(s) of interest simultaneously. This fundamental biological principle has motivated the researchers for developing joint models which can explain the biological system efficiently. Because of the advanced biotechnology, huge amount of biological information can be easily obtained in current years. Hence dimension reduction is one of the major issues in current biological research. In this article, we propose a Bayesian semiparametric approach of jointly modeling observed longitudinal trait and event-time data. A sure independence screening procedure based on the distance correlation and a modified version of Bayesian Lasso are used for dimension reduction. Traditional Cox proportional hazards model is used for modeling the event-time. Our proposed model is used for detecting marker genes controlling the biomass and first flowering time of soybean plants. Simulation studies are performed for assessing the practical usefulness of the proposed model. Proposed model can be used for the joint analysis of traits and diseases for humans, animals and plants.     

Accelerating inference for diffusions observed with measurement error and large sample sizes using approximate Bayesian computation

In recent years, dynamical modelling has been provided with a range of breakthrough methods to perform exact Bayesian inference. However, it is often computationally unfeasible to apply exact statistical methodologies in the context of large data sets and complex models. This paper considers a nonlinear stochastic differential equation model observed with correlated measurement errors and an application to protein folding modelling. An approximate Bayesian computation (ABC)-MCMC algorithm is suggested to allow inference for model parameters within reasonable time constraints. The ABC algorithm uses simulations of ‘subsamples’ from the assumed data-generating model as well as a so-called ‘early-rejection’ strategy to speed up computations in the ABC-MCMC sampler. Using a considerate amount of subsamples does not seem to degrade the quality of the inferential results for the considered applications. A simulation study is conducted to compare our strategy with exact Bayesian inference, the latter resulting two orders of magnitude slower than ABC-MCMC for the considered set-up. Finally, the ABC algorithm is applied to a large size protein data. The suggested methodology is fairly general and not limited to the exemplified model and data.     

A characteristic function-based approach to approximate maximum likelihood estimation

The choice of the summary statistics in approximate maximum likelihood is often a crucial issue. We develop a criterion for choosing the most effective summary statistic and then focus on the empirical characteristic function. In the iid setting, the approximating posterior distribution converges to the approximate distribution of the parameters conditional upon the empirical characteristic function. Simulation experiments suggest that the method is often preferable to numerical maximum likelihood. In a time-series framework, no optimality result can be proved, but the simulations indicate that the method is effective in small samples.     

A Bayesian approach to some overdispersion models

In fitting a generalized linear model, many authors have noticed that data sets can show greater residual variability than predicted under the exponential family. Two main approaches have been used to model this overdispersion. The first approach uses a sampling density which is a conjugate mixture of exponential family distributions. The second uses a quasilikelihood which adds a new scale parameter to the exponential likelihood. The approaches are compared by means of a Bayesian analysis using noninformative priors. In examples, it is indicated that the posterior analysis can be significantly different using the two approaches.     

A nonparametric Bayesian approach to copula estimation

We propose a novel Dirichlet-based Pólya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Furthermore, the continuity of the imposed D-P tree prior leads to a more favourable smoothing effect in copula estimation over classic frequentist methods, especially with small sets of observations. We also apply our method to the copula prediction between the S&P 500 index and the IBM stock prices during the 2007–08 financial crisis, finding that D-P tree-based methods enjoy strong robustness and flexibility over classic methods under such irregular market behaviours.