###### USING AUXILIARY VARIABLE METHOD IN HIERARCHICAL BAYESIAN SPATIOTEMPORAL MODELS, APPLICATION TO AUDITORY fMRI DATA

Urology and Nephrology Research Center, Shahid Beheshti University of Medical Sciences, Tehran, Iran,

Department of Biostatistics, Faculty of Paramedical Sciences, Shahid Beheshti University of Medical Science, Tehran, Iran

Medical Informatics Department, Faculty of Paramedical Sciences, Shahid Beheshti University of Medical Science, Tehran, Iran,

###### ABSTRACT

Sometimes the Bayesian models include a spatial prior which is computationally intractable, because normalizing constants are appeared in the posterior distributions. Computing of normalizing constants is a fundamental computational problem in many Spatiotemporal Bayesian inferences. Functional Magnetic Resonance Imaging (fMRI) data sets are a popular example of huge data sets and big data analytics that their spatial and temporal dependence structures are very complex. Therefore, Spatiotemporal Bayesian inferences for analyzing fMRI data are lionized, but normalizing constant problems often make these models be computationally problematic. In this paper we have focused on the computational schemes for practical Bayesian estimation in binary spatial Ising prior which is one of the problematic priors and is widely used in Spatiotemporal modeling of fMRI data. We investigate the new application of Auxiliary Variable method proposed by Møller for Bayesian estimation in a Hierarchical Spatiotemporal model including an Ising prior, where the posterior involves a normalizing constant. This method avoids approximations such as those in earlier works and also incorporates the normalizing constant and external field problems of Ising prior, simultaneously. We explore the performance of the method on simulated 3D correlated time series. Our approach does a good performance through the simulations. Also, we proceed with real fMRI data set, auditory data from SPM software.

**Keywords:**Auxiliary Variable Method , Binary Spatial Ising , Intractable Normalizing Constant.

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