BAYESIAN SUBSET SELECTION AND MODEL AVERAGING USING A CENTRED AND DISPERSED PRIOR FOR THE ERROR VARIANCE

This article proposes a new data‐based prior distribution for the error variance in a Gaussian linear regression model, when the model is used for Bayesian variable selection and model averaging. For a given subset of variables in the model, this prior has a mode that is an unbiased estimator of the error variance but is suitably dispersed to make it uninformative relative to the marginal likelihood. The advantage of this empirical Bayes prior for the error variance is that it is centred and dispersed sensibly and avoids the arbitrary specification of hyperparameters. The performance of the new prior is compared to that of a prior proposed previously in the literature using several simulated examples and two loss functions. For each example our paper also reports results for the model that orthogonalizes the predictor variables before performing subset selection. A real example is also investigated. The empirical results suggest that for both the simulated and real data, the performance of the estimators based on the prior proposed in our article compares favourably with that of a prior used previously in the literature.     

The ensemble Kalman filter is an ABC algorithm

The ensemble Kalman filter is the method of choice for many difficult high-dimensional filtering problems in meteorology, oceanography, hydrology and other fields. In this note we show that a common variant of the ensemble Kalman filter is an approximate Bayesian computation (ABC) algorithm. This is of interest for a number of reasons. First, the ensemble Kalman filter is an example of an ABC algorithm that predates the development of ABC algorithms. Second, the ensemble Kalman filter is used for very high-dimensional problems, whereas ABC methods are normally applied only in very low-dimensional problems. Third, recent state of the art extensions of the ensemble Kalman filter can also be understood within the ABC framework.     

Likelihood-free approximate Gibbs sampling

Likelihood-free methods such as approximate Bayesian computation (ABC) have extended the reach of statistical inference to problems with computationally intractable likelihoods. Such approaches perform well for small-to-moderate dimensional problems, but suffer a curse of dimensionality in the number of model parameters. We introduce a likelihood-free approximate Gibbs sampler that naturally circumvents the dimensionality issue by focusing on lower-dimensional conditional distributions. These distributions are estimated by flexible regression models either before the sampler is run, or adaptively during sampler implementation. As a result, and in comparison to Metropolis-Hastings-based approaches, we are able to fit substantially more challenging statistical models than would otherwise be possible. We demonstrate the sampler’s performance via two simulated examples, and a real analysis of Airbnb rental prices using a intractable high-dimensional multivariate nonlinear state-space model with a 36-dimensional latent state observed on 365 time points, which presents a real challenge to standard ABC techniques.     

Efficient sampling schemes for Bayesian MARS models with many predictors

Multivariate adaptive regression spline fitting or MARS (Friedman 1991) provides a useful methodology for flexible adaptive regression with many predictors. The MARS methodology produces an estimate of the mean response that is a linear combination of adaptively chosen basis functions. Recently, a Bayesian version of MARS has been proposed (Denison, Mallick and Smith 1998a, Holmes and Denison, 2002) combining the MARS methodology with the benefits of Bayesian methods for accounting for model uncertainty to achieve improvements in predictive performance. In implementation of the Bayesian MARS approach, Markov chain Monte Carlo methods are used for computations, in which at each iteration of the algorithm it is proposed to change the current model by either (a) Adding a basis function (birth step) (b) Deleting a basis function (death step) or (c) Altering an existing basis function (change step). In the algorithm of Denison, Mallick and Smith (1998a), when a birth step is proposed, the type of basis function is determined by simulation from the prior. This works well in problems with a small number of predictors, is simple to program, and leads to a simple form for Metropolis-Hastings acceptance probabilities. However, in problems with very large numbers of predictors where many of the predictors are useless it may be difficult to find interesting interactions with such an approach. In the original MARS algorithm of Friedman (1991) a heuristic is used of building up higher order interactions from lower order ones, which greatly reduces the complexity of the search for good basis functions to add to the model. While we do not exactly follow the intuition of the original MARS algorithm in this paper, we nevertheless suggest a similar idea in which the Metropolis-Hastings proposals of Denison, Mallick and Smith (1998a) are altered to allow dependence on the current model. Our modification allows more rapid identification and exploration of important interactions, especially in problems with very large numbers of predictor variables and many useless predictors. Performance of the algorithms is compared in simulation studies.     

Robust Bayesian synthetic likelihood via a semi-parametric approach

Bayesian synthetic likelihood (BSL) is now a well-established method for performing approximate Bayesian parameter estimation for simulation-based models that do not possess a tractable likelihood function. BSL approximates an intractable likelihood function of a carefully chosen summary statistic at a parameter value with a multivariate normal distribution. The mean and covariance matrix of this normal distribution are estimated from independent simulations of the model. Due to the parametric assumption implicit in BSL, it can be preferred to its nonparametric competitor, approximate Bayesian computation, in certain applications where a high-dimensional summary statistic is of interest. However, despite several successful applications of BSL, its widespread use in scientific fields may be hindered by the strong normality assumption. In this paper, we develop a semi-parametric approach to relax this assumption to an extent and maintain the computational advantages of BSL without any additional tuning. We test our new method, semiBSL, on several challenging examples involving simulated and real data and demonstrate that semiBSL can be significantly more robust than BSL and another approach in the literature.     

Gaussian variational approximation with sparse precision matrices

We consider the problem of learning a Gaussian variational approximation to the posterior distribution for a high-dimensional parameter, where we impose sparsity in the precision matrix to reflect appropriate conditional independence structure in the model. Incorporating sparsity in the precision matrix allows the Gaussian variational distribution to be both flexible and parsimonious, and the sparsity is achieved through parameterization in terms of the Cholesky factor. Efficient stochastic gradient methods that make appropriate use of gradient information for the target distribution are developed for the optimization. We consider alternative estimators of the stochastic gradients, which have lower variation and are more stable. Our approach is illustrated using generalized linear mixed models and state-space models for time series.     

The predictive Lasso

We propose a shrinkage procedure for simultaneous variable selection and estimation in generalized linear models (GLMs) with an explicit predictive motivation. The procedure estimates the coefficients by minimizing the Kullback-Leibler divergence of a set of predictive distributions to the corresponding predictive distributions for the full model, subject to an l ₁ constraint on the coefficient vector. This results in selection of a parsimonious model with similar predictive performance to the full model. Thanks to its similar form to the original Lasso problem for GLMs, our procedure can benefit from available l ₁-regularization path algorithms. Simulation studies and real data examples confirm the efficiency of our method in terms of predictive performance on future observations.     

Variational approximation for heteroscedastic linear models and matching pursuit algorithms

Modern statistical applications involving large data sets have focused attention on statistical methodologies which are both efficient computationally and able to deal with the screening of large numbers of different candidate models. Here we consider computationally efficient variational Bayes approaches to inference in high-dimensional heteroscedastic linear regression, where both the mean and variance are described in terms of linear functions of the predictors and where the number of predictors can be larger than the sample size. We derive a closed form variational lower bound on the log marginal likelihood useful for model selection, and propose a novel fast greedy search algorithm on the model space which makes use of one-step optimization updates to the variational lower bound in the current model for screening large numbers of candidate predictor variables for inclusion/exclusion in a computationally thrifty way. We show that the model search strategy we suggest is related to widely used orthogonal matching pursuit algorithms for model search but yields a framework for potentially extending these algorithms to more complex models. The methodology is applied in simulations and in two real examples involving prediction for food constituents using NIR technology and prediction of disease progression in diabetes.     

Conditionally structured variational Gaussian approximation with importance weights

Statistics and Computing - We develop flexible methods of deriving variational inference for models with complex latent variable structure. By splitting the variables in these models into...     

A general approach to heteroscedastic linear regression

Our article presents a general treatment of the linear regression model, in which the error distribution is modelled nonparametrically and the error variances may be heteroscedastic, thus eliminating the need to transform the dependent variable in many data sets. The mean and variance components of the model may be either parametric or nonparametric, with parsimony achieved through variable selection and model averaging. A Bayesian approach is used for inference with priors that are data-based so that estimation can be carried out automatically with minimal input by the user. A Dirichlet process mixture prior is used to model the error distribution nonparametrically; when there are no regressors in the model, the method reduces to Bayesian density estimation, and we show that in this case the estimator compares favourably with a well-regarded plug-in density estimator. We also consider a method for checking the fit of the full model. The methodology is applied to a number of simulated and real examples and is shown to work well.