Gaussian process hyper-parameter estimation using parallel asymptotically independent Markov sampling

Gaussian process emulators of computationally expensive computer codes provide fast statistical approximations to model physical processes. The training of these surrogates depends on the set of design points chosen to run the simulator. Due to computational cost, such training set is bound to be limited and quantifying the resulting uncertainty in the hyper-parameters of the emulator by uni-modal distributions is likely to induce bias. In order to quantify this uncertainty, this paper proposes a computationally efficient sampler based on an extension of Asymptotically Independent Markov Sampling, a recently developed algorithm for Bayesian inference. Structural uncertainty of the emulator is obtained as a by-product of the Bayesian treatment of the hyper-parameters. Additionally, the user can choose to perform stochastic optimisation to sample from a neighbourhood of the Maximum a Posteriori estimate, even in the presence of multimodality. Model uncertainty is also acknowledged through numerical stabilisation measures by including a nugget term in the formulation of the probability model. The efficiency of the proposed sampler is illustrated in examples where multi-modal distributions are encountered. For the purpose of reproducibility, further development, and use in other applications the code used to generate the examples is freely available for download at https://github.com/agarbuno/paims_codesComment: Computational Statistics \& Data Analysis, Volume 103, November 201

Transitional annealed adaptive slice sampling for Gaussian process hyper-parameter estimation

Surrogate models have become ubiquitous in science and engineering for their capability of emulating expensive computer codes, necessary to model and investigate complex phenomena. Bayesian emulators based on Gaussian processes adequately quantify the uncertainty that results from the cost of the original simulator, and thus the inability to evaluate it on the whole input space. However, it is common in the literature that only a partial Bayesian analysis is carried out, whereby the underlying hyper-parameters are estimated via gradient-free optimization or genetic algorithms, to name a few methods. On the other hand, maximum a posteriori (MAP) estimation could discard important regions of the hyper-parameter space. In this paper, we carry out a more complete Bayesian inference, that combines Slice Sampling with some recently developed sequential Monte Carlo samplers. The resulting algorithm improves the mixing in the sampling through the delayed-rejection nature of Slice Sampling, the inclusion of an annealing scheme akin to Asymptotically Independent Markov Sampling and parallelization via transitional Markov chain Monte Carlo. Examples related to the estimation of Gaussian process hyper-parameters are presented. For the purpose of reproducibility, further development, and use in other applications, the code to generate the examples in this paper is freely available for download at http://github.com/agarbuno/ta2s2_codes

Joint segmentation of multivariate astronomical time series : bayesian sampling with a hierarchical model

Astronomy and other sciences often face the problem of detecting and characterizing structure in two or more related time series. This paper approaches such problems using Bayesian priors to represent relationships between signals with various degrees of certainty, and not just rigid constraints. The segmentation is conducted by using a hierarchical Bayesian approach to a piecewise constant Poisson rate model. A Gibbs sampling strategy allows joint estimation of the unknown parameters and hyperparameters. Results obtained with synthetic and real photon counting data illustrate the performance of the proposed algorithm

Collaborative sparse regression using spatially correlated supports - Application to hyperspectral unmixing

This paper presents a new Bayesian collaborative sparse regression method for linear unmixing of hyperspectral images. Our contribution is twofold; first, we propose a new Bayesian model for structured sparse regression in which the supports of the sparse abundance vectors are a priori spatially correlated across pixels (i.e., materials are spatially organised rather than randomly distributed at a pixel level). This prior information is encoded in the model through a truncated multivariate Ising Markov random field, which also takes into consideration the facts that pixels cannot be empty (i.e, there is at least one material present in each pixel), and that different materials may exhibit different degrees of spatial regularity. Secondly, we propose an advanced Markov chain Monte Carlo algorithm to estimate the posterior probabilities that materials are present or absent in each pixel, and, conditionally to the maximum marginal a posteriori configuration of the support, compute the MMSE estimates of the abundance vectors. A remarkable property of this algorithm is that it self-adjusts the values of the parameters of the Markov random field, thus relieving practitioners from setting regularisation parameters by cross-validation. The performance of the proposed methodology is finally demonstrated through a series of experiments with synthetic and real data and comparisons with other algorithms from the literature

A Bayesian Poisson-Gaussian Process Model for Popularity Learning in Edge-Caching Networks

Edge-caching is recognized as an efficient technique for future cellular networks to improve network capacity and user-perceived quality of experience. To enhance the performance of caching systems, designing an accurate content request prediction algorithm plays an important role. In this paper, we develop a flexible model, a Poisson regressor based on a Gaussian process, for the content request distribution. The first important advantage of the proposed model is that it encourages the already existing or seen contents with similar features to be correlated in the feature space and therefore it acts as a regularizer for the estimation. Second, it allows to predict the popularities of newly-added or unseen contents whose statistical data is not available in advance. In order to learn the model parameters, which yield the Poisson arrival rates or alternatively the content \textit{popularities}, we invoke the Bayesian approach which is robust against over-fitting. However, the resulting posterior distribution is analytically intractable to compute. To tackle this, we apply a Markov Chain Monte Carlo (MCMC) method to approximate this distribution which is also asymptotically exact. Nevertheless, the MCMC is computationally demanding especially when the number of contents is large. Thus, we employ the Variational Bayes (VB) method as an alternative low complexity solution. More specifically, the VB method addresses the approximation of the posterior distribution through an optimization problem. Subsequently, we present a fast block-coordinate descent algorithm to solve this optimization problem. Finally, extensive simulation results both on synthetic and real-world datasets are provided to show the accuracy of our prediction algorithm and the cache hit ratio (CHR) gain compared to existing methods from the literature