Latent Gaussian modeling and INLA: A review with focus on space-time applications

Bayesian hierarchical models with latent Gaussian layers have proven very flexible in capturing complex stochastic behavior and hierarchical structures in high-dimensional spatial and spatio-temporal data. Whereas simulation-based Bayesian inference through Markov Chain Monte Carlo may be hampered by slow convergence and numerical instabilities, the inferential framework of Integrated Nested Laplace Approximation (INLA) is capable to provide accurate and relatively fast analytical approximations to posterior quantities of interest. It heavily relies on the use of Gauss-Markov dependence structures to avoid the numerical bottleneck of high-dimensional nonsparse matrix computations. With a view towards space-time applications, we here review the principal theoretical concepts, model classes and inference tools within the INLA framework. Important elements to construct space-time models are certain spatial Mat\'ern-like Gauss-Markov random fields, obtained as approximate solutions to a stochastic partial differential equation. Efficient implementation of statistical inference tools for a large variety of models is available through the INLA package of the R software. To showcase the practical use of R-INLA and to illustrate its principal commands and syntax, a comprehensive simulation experiment is presented using simulated non Gaussian space-time count data with a first-order autoregressive dependence structure in time

Variational semi-blind sparse deconvolution with orthogonal kernel bases and its application to MRFM

We present a variational Bayesian method of joint image reconstruction and point spread function (PSF) estimation when the PSF of the imaging device is only partially known. To solve this semi-blind deconvolution problem, prior distributions are specified for the PSF and the 3D image. Joint image reconstruction and PSF estimation is then performed within a Bayesian framework, using a variational algorithm to estimate the posterior distribution. The image prior distribution imposes an explicit atomic measure that corresponds to image sparsity. Importantly, the proposed Bayesian deconvolution algorithm does not require hand tuning. Simulation results clearly demonstrate that the semi-blind deconvolution algorithm compares favorably with previous Markov chain Monte Carlo (MCMC) version of myopic sparse reconstruction. It significantly outperforms mismatched non-blind algorithms that rely on the assumption of the perfect knowledge of the PSF. The algorithm is illustrated on real data from magnetic resonance force microscopy (MRFM)

Variational semi-blind sparse deconvolution with orthogonal kernel bases and its application to MRFM

We present a variational Bayesian method of joint image reconstruction and point spread function (PSF) estimation when the PSF of the imaging device is only partially known. To solve this semi-blind deconvolution problem, prior distributions are specified for the PSF and the 3D image. Joint image reconstruction and PSF estimation is then performed within a Bayesian framework, using a variational algorithm to estimate the posterior distribution. The image prior distribution imposes an explicit atomic measure that corresponds to image sparsity. Importantly, the proposed Bayesian deconvolution algorithm does not require hand tuning. Simulation results clearly demonstrate that the semi-blind deconvolution algorithm compares favorably with previous Markov chain Monte Carlo (MCMC) version of myopic sparse reconstruction. It significantly outperforms mismatched non-blind algorithms that rely on the assumption of the perfect knowledge of the PSF. The algorithm is illustrated on real data from magnetic resonance force microscopy (MRFM)

On-line adaptive learning of the correlated continuous density hidden Markov models for speech recognition

We extend our previously proposed quasi-Bayes adaptive learning framework to cope with the correlated continuous density hidden Markov models (HMMs) with Gaussian mixture state observation densities in which all mean vectors are assumed to be correlated and have a joint prior distribution. A successive approximation algorithm is proposed to implement the correlated mean vectors' updating. As an example, by applying the method to an on-line speaker adaptation application, the algorithm is experimentally shown to be asymptotically convergent as well as being able to enhance the efficiency and the effectiveness of the Bayes learning by taking into account the correlation information between different model parameters. The technique can be used to cope with the time-varying nature of some acoustic and environmental variabilities, including mismatches caused by changing speakers, channels, transducers, environments, and so on.published_or_final_versio

Type Ia Supernova Light Curve Inference: Hierarchical Bayesian Analysis in the Near Infrared

We present a comprehensive statistical analysis of the properties of Type Ia SN light curves in the near infrared using recent data from PAIRITEL and the literature. We construct a hierarchical Bayesian framework, incorporating several uncertainties including photometric error, peculiar velocities, dust extinction and intrinsic variations, for coherent statistical inference. SN Ia light curve inferences are drawn from the global posterior probability of parameters describing both individual supernovae and the population conditioned on the entire SN Ia NIR dataset. The logical structure of the hierarchical model is represented by a directed acyclic graph. Fully Bayesian analysis of the model and data is enabled by an efficient MCMC algorithm exploiting the conditional structure using Gibbs sampling. We apply this framework to the JHK_s SN Ia light curve data. A new light curve model captures the observed J-band light curve shape variations. The intrinsic variances in peak absolute magnitudes are: sigma(M_J) = 0.17 +/- 0.03, sigma(M_H) = 0.11 +/- 0.03, and sigma(M_Ks) = 0.19 +/- 0.04. We describe the first quantitative evidence for correlations between the NIR absolute magnitudes and J-band light curve shapes, and demonstrate their utility for distance estimation. The average residual in the Hubble diagram for the training set SN at cz > 2000 km/s is 0.10 mag. The new application of bootstrap cross-validation to SN Ia light curve inference tests the sensitivity of the model fit to the finite sample and estimates the prediction error at 0.15 mag. These results demonstrate that SN Ia NIR light curves are as effective as optical light curves, and, because they are less vulnerable to dust absorption, they have great potential as precise and accurate cosmological distance indicators.Comment: 24 pages, 15 figures, 4 tables. Accepted for publication in ApJ. Corrected typo, added references, minor edit

Simultaneous Image Restoration and Hyperparameter Estimation for Incomplete Data by a Cumulant Analysis

The purpose of this report is first to show the main properties of Gibbs distributions considered as exponential statistics on finite spaces, as well as their sampling and annealing properties. Moreover, the definition and use of their cumulant expansions enables to exhibit other important properties of such distributions. Last, we tackle the problem of hyperparameter estimation in an incomplete data frame for image restoration purposes. A detailed analysis of several joint restoration-estimation methods using generalized stochastic gradient algorithms is presented, requiring infinite, continuous configuration spaces. Using once again cumulant analysis and its relationship with Statistical Physics allows us to propose new algorithms and to extend them to an explicit boundary frame