Likelihood-free Bayesian estimation of multivariate quantile distributions

Modeling sign concordance of quantile regression residuals with multiple outcomes

Quantile regression permits describing how quantiles of a scalar response variable depend on a set of predictors. Because a unique definition of multivariate quantiles is lacking, extending quantile regression to multivariate responses is somewhat complicated. In this paper, we describe a simple approach based on a two-step procedure: in the first step, quantile regression is applied to each response separately; in the second step, the joint distribution of the signs of the residuals is modeled through multinomial regression. The described approach does not require a multidimensional definition of quantiles, and can be used to capture important features of a multivariate response and assess the effects of covariates on the correlation structure. We apply the proposed method to analyze two different datasets.

A Global Stochastic Optimization Particle Filter Algorithm

We introduce a new online algorithm for expected loglikelihood maximization in situations where the objective function is multi-modal and/or has saddle points. The key element underpinning the algorithm is a probability distribution which (a) is shown to concentrate on the target parameter value as the sample size increases and (b) can be efficiently estimated by means of a standard particle filter algorithm. This distribution depends on a learning rate, where the faster the learning rate the quicker it concentrates on the desired element of the search space, but the less likely the algorithm is to escape from a local optimum of the objective function. In order to achieve a fast convergence rate with a slow learning rate, our algorithm exploits the acceleration property of averaging, well-known in the stochastic gradient literature. Considering several challenging estimation problems, the numerical experiments show that, with high probability, the algorithm successfully finds the highest mode of the objective function and converges to its global maximizer at the optimal rate. While the focus of this work is expected loglikelihood maximization, the proposed methodology and its theory apply more generally for optimizing a function defined through an expectation.

Bayesian computation via empirical likelihood

Approximate Bayesian computation 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 approximate Bayesian computation 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 Bayesian computation with empirical likelihood 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.