Non-linear regression models for Approximate Bayesian Computation

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 importance sampling. The new algorithm is compared to the state-of-the-art approximate Bayesian methods, and achieves considerable reduction of the computational burden in two examples of inference in statistical genetics and in a queueing model.Comment: 4 figures; version 3 minor changes; to appear in Statistics and Computin

Bayesian computation via empirical likelihood

Approximate Bayesian computation (ABC) has become an essential tool for the analysis of complex stochastic models when the likelihood function is numerically unavailable. However, the well-established statistical method of empirical likelihood provides another route to such settings that bypasses simulations from the model and the choices of the ABC parameters (summary statistics, distance, tolerance), while being convergent in the number of observations. Furthermore, bypassing model simulations may lead to significant time savings in complex models, for instance those found in population genetics. The BCel algorithm we develop in this paper also provides an evaluation of its own performance through an associated effective sample size. The method is illustrated using several examples, including estimation of standard distributions, time series, and population genetics models.Comment: 21 pages, 12 figures, revised version of the previous version with a new titl

A subsampled double bootstrap for massive data

The bootstrap is a popular and powerful method for assessing precision of estimators and inferential methods. However, for massive datasets which are increasingly prevalent, the bootstrap becomes prohibitively costly in computation and its feasibility is questionable even with modern parallel computing platforms. Recently Kleiner, Talwalkar, Sarkar, and Jordan (2014) proposed a method called BLB (Bag of Little Bootstraps) for massive data which is more computationally scalable with little sacrifice of statistical accuracy. Building on BLB and the idea of fast double bootstrap, we propose a new resampling method, the subsampled double bootstrap, for both independent data and time series data. We establish consistency of the subsampled double bootstrap under mild conditions for both independent and dependent cases. Methodologically, the subsampled double bootstrap is superior to BLB in terms of running time, more sample coverage and automatic implementation with less tuning parameters for a given time budget. Its advantage relative to BLB and bootstrap is also demonstrated in numerical simulations and a data illustration

ABC likelihood-freee methods for model choice in Gibbs random fields

Gibbs random fields (GRF) are polymorphous statistical models that can be used to analyse different types of dependence, in particular for spatially correlated data. However, when those models are faced with the challenge of selecting a dependence structure from many, the use of standard model choice methods is hampered by the unavailability of the normalising constant in the Gibbs likelihood. In particular, from a Bayesian perspective, the computation of the posterior probabilities of the models under competition requires special likelihood-free simulation techniques like the Approximate Bayesian Computation (ABC) algorithm that is intensively used in population genetics. We show in this paper how to implement an ABC algorithm geared towards model choice in the general setting of Gibbs random fields, demonstrating in particular that there exists a sufficient statistic across models. The accuracy of the approximation to the posterior probabilities can be further improved by importance sampling on the distribution of the models. The practical aspects of the method are detailed through two applications, the test of an iid Bernoulli model versus a first-order Markov chain, and the choice of a folding structure for two proteins.Comment: 19 pages, 5 figures, to appear in Bayesian Analysi