Bayesian Analysis

Bayesian Spatiotemporal Modeling Using Hierarchical Spatial Priors, with Applications to Functional Magnetic Resonance Imaging (with Discussion)

Martin Bezener, John Hughes, and Galin Jones

Full-text: Open access


We propose a spatiotemporal Bayesian variable selection model for detecting activation in functional magnetic resonance imaging (fMRI) settings. Following recent research in this area, we use binary indicator variables for classifying active voxels. We assume that the spatial dependence in the images can be accommodated by applying an areal model to parcels of voxels. The use of parcellation and a spatial hierarchical prior (instead of the popular Ising prior) results in a posterior distribution amenable to exploration with an efficient Markov chain Monte Carlo (MCMC) algorithm. We study the properties of our approach by applying it to simulated data and an fMRI data set.

Article information

Bayesian Anal., Volume 13, Number 4 (2018), 1261-1313.

First available in Project Euclid: 8 May 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bayesian variable selection fMRI MCMC spatiotemporal areal model

Creative Commons Attribution 4.0 International License.


Bezener, Martin; Hughes, John; Jones, Galin. Bayesian Spatiotemporal Modeling Using Hierarchical Spatial Priors, with Applications to Functional Magnetic Resonance Imaging (with Discussion). Bayesian Anal. 13 (2018), no. 4, 1261--1313. doi:10.1214/18-BA1108.

Export citation


  • Banerjee, S., Carlin, B. P., and Gelfand, A. E. (2003). Hierarchical Modeling and Analysis for Spatial Data. New York: Chapman and Hall/CRC, 1st edition.
  • Bezener, M., Hughes, J., and Jones, G. (2018). “Supplemental Material for “Bayesian Spatiotemporal Modeling using Hierarchical Spatial Priors, with Applications to Functional Magnetic Resonance Imaging”.” Bayesian Analysis.
  • Blei, D. M., Kucukelbir, A., and McAuliffe, J. D. (2017). “Variational inference: a review for statisticians.” Journal of the American Statistical Association, 112: 859–877.
  • Bowman, F. D. (2014). “Brain Imaging Analysis.” Annual Review of Statistics and Its Application, 1: 61–85.
  • Bowman, F. D., Caffo, B., Bassett, S. S., and Kilts, C. (2007). “A Bayesian Hierarchical Framework for spatial modeling of fMRI data.” NeuroImage, 39: 146–156.
  • Cipra, B. (1987). “An Introduction to the Ising Model.” American Mathematical Monthly, 94: 937–959.
  • Cressie, N. A. (1993). Statistics for Spatial Data. New York: Wiley Interscience, Revised edition.
  • Essen, D. C. V., Smith, S. M., Barch, D. M., Behrens, T. E., Yavoub, E., and Ugurbil, K. (2013). “The WU-Minn Human Connectome Project: An overview.” NeuroImage, 62–79.
  • Flegal, J. M., Haran, M., and Jones, G. L. (2008). “Markov Chain Monte Carlo: Can We Trust the Third Significant Figure?” Statistical Science, 23: 250–260.
  • Flegal, J. M., Hughes, J., Vats, D., and Dai, N. (2017). mcmcse: Monte Carlo Standard Errors for MCMC. Riverside, CA, Denver, CO, Coventry, UK, and Minneapolis, MN. R package version 1.3–2.
  • Friston, K. J., Ashburner, J. T., Kiebel, S. J., Nichols, T. E., and Penny, W. D. (2007). Statistical Parametric Mapping: The Analysis of Functional Brain Images. London: Academic Press.
  • Friston, K. J., Holmes, A., Worsley, K. J., Polin, J. B., Frith, C., and Frackowik, R. (1995). “Statistical parametric maps in functional imaging: A general linear approach.” Human Brain Mapping, 2: 189–210.
  • Friston, K. J., Worsley, K., Frackowiak, R., Mazziotta, J., and Evans, A. (1994). “Assessing the significance of focal activations using their spatial extent.” Human Brain Mapping, 1: 210–220.
  • Genovese, C. R. (2000). “A Bayesian Time-Course Model for Functional Magnetic Resonance Imaging Data.” Journal of the American Statistical Association, 95: 691–703.
  • George, E. I. and McCulloch, R. E. (1993). “Variable Selection Via Gibbs Sampling.” Journal of the American Statistical Association, 88: 881–889.
  • George, E. I. and McCulloch, R. E. (1997). “Approaches for Bayesian Variable Selection.” Statistica Sinica, 7: 339–373.
  • Gössel, C., Auer, D., and Fahrmeir, L. (2001). “Bayesian Spatiotemporal Inference in Functional Magnetic Resonance Imaging.” Biometrics, 57: 554–562.
  • Haran, M. (2011). “Gaussian random field models for spatial data.” In Brooks, S. P., Gelman, A. E., Jones, G. L., and Meng, X. L. (eds.), Handbook of Markov Chain Monte Carlo, 449–478. London: Chapman and Hall/CRC.
  • Hariri, A. R., Mattay, V. S., Tessitore, A., Kolachana, B., Fera, F., Goldman, D., Egan, M. F., and Weinberger, D. R. (2002). “Serotonin Transporter Genetic Variation and the Response of Human Amygdala.” Science, 297: 400–4003.
  • Huettel, S. A., Somng, A. W., and McCarthy, G. (2009). Functional Magnetic Resonance Imaging. Sunderland, MA: Sinauer Associates.
  • Jones, G. L., Haran, M., Caffo, B. S., and Neath, R. (2006). “Fixed-width output analysis for Markov chain Monte Carlo.” Journal of the American Statistical Association, 101: 1537–1547.
  • Käll, L., Storey, J. D., MacCoss, M. J., and Noble, W. S. (2008). “Posterior error probabilities and false discovery rates: two sides of the same coin.” Journal of Proteome Research, 7: 40–44.
  • Kaushik, K., Karesh, K., and Suresha, D. (2013). “Segmentation of the white matter from the brain fMRI images.” International Journal of Advanced Research in Computer Engineering and Technology, 2: 1314–1317.
  • Landman, B. A., Yang, X., and Kang, H. (2012). “Do we really need robust and altrernative inference methods for brain MRI?” In Yap, P., Liu, T., Shen, D., and Westin, C. (eds.), MBIA 2012: Multimodal Brain Image Analysis, volume 7509 of Lecture Notes in Computer Science, 77–93. Berlin: Springer.
  • Lazar, N. A. (2008). The Statistical Analysis of fMRI Data. New York: Springer.
  • Lee, K.-J., Jones, G. L., Caffo, B. S., and Bassett, S. S. (2014). “Spatial Bayesian Variable Selection Models on Functional Magnetic Resonance Imaging Time-Series Data.” Bayesian Analysis, 9: 699–732.
  • Lindquist, M. A. (2008). “The Statistical Analysis of fMRI Data.” Statistical Science, 23: 439–464.
  • Locascio, J., Jennings, P. J., Moore, C. I., and Corkin, S. (1997). “Time series analysis in the time domain and resampling methods for studies of functional magnetic brain imaging.” Human Brain Mapping, 168–193.
  • Makni, S., Idier, J., Vincent, T., Thirion, B., Dehaene-Lambertz, G., and Ciuciu, P. (2008). “A fully Bayesian approach to the parcel-based detection-estimation of brain activity in fMRI.” NeuroImage, 41: 941–969.
  • Mikl, M., Mareček, R., Hluštík, P., Pavlicová, M., Drastich, A., Chlebus, P., Brázdil, M., and Krupa, P. (2008). “Effects of spatial smoothing on fMRI group inferences.” Magnetic Resonance Imaging, 26: 490–503.
  • Monti, M. M. (2011). “Statistical analysis of fMRI time-series: A critical review of the GLM approach.” Frontiers in Human Neuroscience, 5.
  • Morris, R., Descombes, X., and Zerubia, J. (1996). “The Ising/Potts model is not well suited to segmentation tasks.” In Digital Signal Processing Workshop Proceedings, 263–265. IEEE.
  • Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective. Cambridge: The MIT Press.
  • Musgrove, D. R., Hughes, J., and Eberly, L. E. (2016). “Fast, fully Bayesian spatiotemporal inference for fMRI data.” Biostatistics, 17: 291–303.
  • Penny, W., Kiebel, S., and Friston, K. (2003). “Variational Bayesian inference for fMRI time series.” NeuroImage, 19: 727–741.
  • Penny, W. D., Trujillo-Barreto, N. J., and Friston, K. J. (2005). “Bayesian fMRI time series analysis with spatial priors.” NeuroImage, 24: 350–362.
  • Quirós, A., Diez, R. M., and Wilson, S. P. (2010). “Bayesian spatiotemporal model of fMRI data using transfer functions.” NeuroImage, 52: 9995–1004.
  • Raftery, A. (1996). “Hypothesis Testing and Model Selection.” In Gilks, W., Spiegelhalter, D., and Richardson, S. (eds.), Markov Chain Monte Carlo in Practice. London: Chapman and Hall.
  • Smith, D. and Smith, M. (2006). “Estimation of Binary Markov Random Fields Using Markov Chain Monte Carlo.” Journal of Computational and Graphical Statistics, 15: 207–227.
  • Smith, M. and Fahrmeir, L. (2007). “Spatial Bayesian Variable Selection with Application to Functional Magnetic Resonance Imaging.” Journal of the American Statistical Association, 102: 417–431.
  • Smith, M. and Kohn, R. (1996). “Nonparametric regression using Bayesian variable selection.” Econometrics, 75: 317–343.
  • Smith, M., Pütz, B., Auer, D., and Fahrmeir, L. (2003). “Assessing brain activity through spatial Bayesian variable selection.” NeuroImage, 20.
  • Storey, J. D. (2003). “The positive false discovery rate: a Bayesian interpretation and the $q$-value.” The Annals of Statistics, 31: 2013–2035.
  • Triantafyllou, C., Hoge, R., and Wald, L. (2006). “Effect of spatial smoothing on physiological noise in high-resolution fMRI.” NeuroImage, 32: 551–557.
  • Vats, D., Flegal, J. M., and Jones, G. L. (2016). “Multivariate output analysis for Markov chain Monte Carlo.” Preprint arXiv:1512.07713.
  • Woolrich, M. W., Jenkinson, M., Brady, J. M., and Smith, S. M. (2004). “Fully Bayesian Spatio-Temporal Modeling of fMRI Data.” IEEE Transactions on Medical Imaging, 23: 213–231.
  • Worsley, K. (2003). “Detecting activation in fMRI data.” Statistical Methods in Medical Research, 12: 401–418.
  • Worsley, K., Marrett, S., Neelin, P., and Evans, A. (1992). “A three-dimensional statistical analysis for CBF activation studies in human brain.” Journal of Cerebral Blood Flow and Metabolism, 12: 900–918.
  • Worsley, K. J., Liao, C. H., Aston, J., Petre, V., Duncan, G. H., Morales, F., and Evans, A. C. (2002). “A General Statistical Analysis for fMRI Data.” NeuroImage, 15: 1–15.
  • Xia, J., Liang, F., and Wang, Y. M. (2009a). “FMRI analysis through Bayesian variable selection with a spatial prior.” In Proceedings of the 6th IEEE International Symposium on Biomedical Imaging, 714–717. IEEE.
  • Xia, J., Liang, F., and Wang, Y. M. (2009b). “fMRI analysis through Bayesian variable selection with a spatial prior.” IEEE Int. Symp. on Biomedical Imaging (ISBI), 714–717.
  • Zellner, A. (1996). “On assessing prior distributions and Bayesian regression analysis with $g$-prior distributions.” In Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti North-Holland/Elsevier, 233–243.
  • Zhang, L., Guindani, M., and Vannucci, M. (2015). “Bayesian models for functional magnetic resonance imaging data analysis.” WIREs Computational Statistics, 7: 21–41.
  • Zhang, L., Guindani, M., Versace, F., Engelmann, J. M., and Vannucci, M. (2016). “A spatio-temporal nonparametric Bayesian variable selection model of multi-subject fMRI data.” The Annals of Applied Statistics, 10: 638–666.
  • Zhang, L., Guindani, M., Versace, F., and Vannucci, M. (2014). “A spatio-temporal nonparametric Bayesian variable selection model of fMRI data for clustering correlated time courses.” NeuroImage, 95: 162–175.
  • Zhou, X. and Schmidler, S. C. (2009). “Bayesian Parameter Estimation in Ising and Potts Models: A Comparative Study with Applications to Protein Modeling.” Technical report, Duke University.

Supplemental materials