Hyper-g Priors for Generalized Linear Models

We develop an extension of the classical Zellner's g-prior to generalized linear models. The prior on the hyperparameter g is handled in a flexible way, so that any continuous proper hyperprior f(g) can be used, giving rise to a large class of hyper-g priors. Connections with the literature are described in detail. A fast and accurate integrated Laplace approximation of the marginal likelihood makes inference in large model spaces feasible. For posterior parameter estimation we propose an efficient and tuning-free Metropolis-Hastings sampler. The methodology is illustrated with variable selection and automatic covariate transformation in the Pima Indians diabetes data set.Comment: 30 pages, 12 figures, poster contribution at ISBA 201

Semi-automatic selection of summary statistics for ABC model choice

A central statistical goal is to choose between alternative explanatory models of data. In many modern applications, such as population genetics, it is not possible to apply standard methods based on evaluating the likelihood functions of the models, as these are numerically intractable. Approximate Bayesian computation (ABC) is a commonly used alternative for such situations. ABC simulates data x for many parameter values under each model, which is compared to the observed data xobs. More weight is placed on models under which S(x) is close to S(xobs), where S maps data to a vector of summary statistics. Previous work has shown the choice of S is crucial to the efficiency and accuracy of ABC. This paper provides a method to select good summary statistics for model choice. It uses a preliminary step, simulating many x values from all models and fitting regressions to this with the model as response. The resulting model weight estimators are used as S in an ABC analysis. Theoretical results are given to justify this as approximating low dimensional sufficient statistics. A substantive application is presented: choosing between competing coalescent models of demographic growth for Campylobacter jejuni in New Zealand using multi-locus sequence typing data

Approximate Bayesian Computation by Modelling Summary Statistics in a Quasi-likelihood Framework

Approximate Bayesian Computation (ABC) is a useful class of methods for Bayesian inference when the likelihood function is computationally intractable. In practice, the basic ABC algorithm may be inefficient in the presence of discrepancy between prior and posterior. Therefore, more elaborate methods, such as ABC with the Markov chain Monte Carlo algorithm (ABC-MCMC), should be used. However, the elaboration of a proposal density for MCMC is a sensitive issue and very difficult in the ABC setting, where the likelihood is intractable. We discuss an automatic proposal distribution useful for ABC-MCMC algorithms. This proposal is inspired by the theory of quasi-likelihood (QL) functions and is obtained by modelling the distribution of the summary statistics as a function of the parameters. Essentially, given a real-valued vector of summary statistics, we reparametrize the model by means of a regression function of the statistics on parameters, obtained by sampling from the original model in a pilot-run simulation study. The QL theory is well established for a scalar parameter, and it is shown that when the conditional variance of the summary statistic is assumed constant, the QL has a closed-form normal density. This idea of constructing proposal distributions is extended to non constant variance and to real-valued parameter vectors. The method is illustrated by several examples and by an application to a real problem in population genetics.Comment: Published at http://dx.doi.org/10.1214/14-BA921 in the Bayesian Analysis (http://projecteuclid.org/euclid.ba) by the International Society of Bayesian Analysis (http://bayesian.org/