Méthodes bayésiennes variationnelles : concepts et applications en neuroimagerie
Journal de la société française de statistique, Tome 151 (2010) no. 2, pp. 107-131.

En estimation bayésienne, les lois a posteriori sont rarement accessibles, même par des méthodes de Monte-Carlo par Chaîne de Markov. Les méthodes bayésiennes variationnelles permettent de calculer directement (et rapidement) une approximation déterministe des lois a posteriori. Cet article décrit le principe des méthodes variationnelles et leur application à l’inférence bayésienne, fait le point sur les principaux résultats théoriques et présente deux exemples d’utilisation en neuroimagerie.

Bayesian posterior distributions can be numerically intractable, even by the means of Markov Chains Monte Carlo methods. Bayesian variational methods can then be used to compute directly (and fast) a deterministic approximation of these posterior distributions. This paper describes the principle of variational methods and their applications in the Bayesian inference, surveys the main theoretical results and details two examples in the neuroimage field.

Mot clés : Méthodes variationnelles, analyse bayésienne, approximation en champ moyen, approximation de Laplace, algorithme EM, données IRMf
Keywords: Variational methods, Bayesian analysis, mean field approximation, Laplace approximation, EM algorithm, IRMf data 
@article{JSFS_2010__151_2_107_0,
     author = {Keribin, Christine},
     title = {M\'ethodes bay\'esiennes variationnelles~: concepts et applications en neuroimagerie},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {107--131},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {151},
     number = {2},
     year = {2010},
     mrnumber = {2775590},
     zbl = {1316.62041},
     language = {fr},
     url = {http://archive.numdam.org/item/JSFS_2010__151_2_107_0/}
}
TY  - JOUR
AU  - Keribin, Christine
TI  - Méthodes bayésiennes variationnelles : concepts et applications en neuroimagerie
JO  - Journal de la société française de statistique
PY  - 2010
SP  - 107
EP  - 131
VL  - 151
IS  - 2
PB  - Société française de statistique
UR  - http://archive.numdam.org/item/JSFS_2010__151_2_107_0/
LA  - fr
ID  - JSFS_2010__151_2_107_0
ER  - 
%0 Journal Article
%A Keribin, Christine
%T Méthodes bayésiennes variationnelles : concepts et applications en neuroimagerie
%J Journal de la société française de statistique
%D 2010
%P 107-131
%V 151
%N 2
%I Société française de statistique
%U http://archive.numdam.org/item/JSFS_2010__151_2_107_0/
%G fr
%F JSFS_2010__151_2_107_0
Keribin, Christine. Méthodes bayésiennes variationnelles : concepts et applications en neuroimagerie. Journal de la société française de statistique, Tome 151 (2010) no. 2, pp. 107-131. http://archive.numdam.org/item/JSFS_2010__151_2_107_0/

[1] Anderson, T.W. An introduction to multivariate Statistical Analysis, John Wiley and Sons, 2003 | MR

[2] Atias, H. Inferring parameters ans Structure of Latent Variable Models by Variational Bayes, Proc. 15th Conference of Uncertainty in Artificial Intelligence (1999), pp. 21-30

[3] Beal, M. J. Variational algorithms for approximate bayesian inference, University College London (2003) (Ph. D. Thesis)

[4] Beal, M. J.; Ghahramani, Z. Variational Bayesian Learning of Directed Graphical Models with Hidden Variables, Bayesian Analysis, Volume 1 (2004) no. 4, pp. 793-832 | DOI | MR | Zbl

[5] Bishop, C. M. Pattern Recognition and Machine Learning, Springer, 2008 | MR | Zbl

[6] Consonni, G.; Marin, J-M. Mean-field variational approximate Bayesian inference for latent variable models, Computational Statistics & Data Analysis, Volume 52 (2007), pp. 790-798 | DOI | MR | Zbl

[7] Dempster, A.P.; Laird, N.M.; Rubin, D.B. Maximum Likelihood from Incomplete Data via the EM Algorithm, Journal of the Royal Statistical Society. Series B (Methodological), Volume 39 (1977) no. 1, pp. 1-38 | DOI | MR | Zbl

[8] Friston, K.; Chu, C.; Mourão-Miranda, J; Hulme, O.; Rees, G.; Penny, W.; Ashburner, J. Bayesian decoding of brain images, NeuroImage, Volume 39 (2008), pp. 181-205 | DOI

[9] Frackowiak, R. S. J.; Friston, K.J.; Frith, C.; Dolan, R.; Price, C.; Ashburner, J.; Penny, W.; Zeki, S. Human Brain Function, Academic Press, 2003

[10] Friston, K.; Mattout, J.; Trujillo-Barreto, N.; Ashburner, J.; W., Penny Variational free energy and the Laplace approximation, NeuroImage, Volume 34 (2006), pp. 220-234 | DOI

[11] Hall, P.; Humphreys, K.; Titterington, D. M. On the adequacy of variational lower bound functions for likelihood-based inference in Markovian models with missing values, J. R. Statist. Soc B, Volume 64 (2002) no. 3, pp. 549-564 | DOI | MR | Zbl

[12] Humphreys, K.; Titterington, D. M. Approximate Bayesian inference for simple mixtures, Proc. Computational Satistics, Physica-Verlag (2000), pp. 331-336 | Zbl

[13] Jordan, M. I. Graphical Models, Statistical Science, Volume 19 (2004) no. 1, pp. 140-155 | DOI | MR | Zbl

[14] Lee, K.; Mengersen, K.L.; Marin, J.-M.; Robert, C.P Bayesian Inference on Mixtures of Distributions, Proceedings of the Platinum Jubilee of the Indian Statistical Institute (2008) | MR

[15] Minka, T. Expectation propagation and approximate Bayesian Inference, Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence, Morgan Kaufmann, San Francisco, CA (2001), pp. 362-369

[16] Rue, H.; Martino, S.; Chopin, N. Approximate Bayesian inference for latent Gaussian models by using intergated nested Laplace approximations, J. R. Statist. Soc. B, Volume 71 (2009) no. 2, pp. 1-35 | MR | Zbl

[17] Robert, C.P. Le choix bayésien : Principes et pratiques, Springer, 2006

[18] Titterington, D. M. Bayesian Methods for Neural Networks and Related Methods, Statistical Science, Volume 19 (2004) no. 1, pp. 128-139 | DOI | MR | Zbl

[19] Tierney, L.; Kadane, J. B. Accurate Approximations for Posterior Moments and Marginal Densities, Journal of the American Statistical Association, Volume 81 (1986), pp. 82-86 | DOI | MR | Zbl

[20] Woolrich, M. W.; Behrens Variational Bayes Inference for Spatial Mixture Models for Segmentation, IEEE Trans. Med. Imag., Volume 25 (2006) no. 10 | DOI

[21] Woolrich, M. W.; Behrens, T. E.; Beckmann, C. F.; Smith, S. M. Mixture Models with Adaptative Spatial Regularisation for Segmentation with application to FMRI Data, IEEE Trans. Med. Imag., Volume 24 (2005) no. 1, pp. 1-11 | DOI

[22] Wainwright, M. J.; Jordan, M. I. Graphical Models, Exponential Families, and Variational Inference, Foundations and trends in Machine Learning, Volume 1 (2008) no. 1-2, pp. 1-305 | Zbl

[23] Worsley, K.J.; Liao, C.H.; Aston, J.; Petre, V.; Duncan, G.H.; Morales, F.; Evans, A.C. A general statistical analysis for fMRI data, Neuroimage, Volume 15 (2002) no. 1, pp. 1-15 | DOI

[24] Wang, B.; Titterington, D. M. Convergence and asymptotique normality of vatiational Bayesian approximations for exponential familty models with missing values, Proceeding of the twentieth conference on uncertainty and artificial interlligence, AUAJ Press, Banff, Canada (2004), pp. 577-584

[25] Wang, B.; Titterington, D. M. Lack of consistency of mean field and variational Bayes approximations for state space models, Neural Proceesing Letters, Volume 20 (2004) no. 3, pp. 151-170 | DOI

[26] Wang, B.; Titterington, D. M. Inadequacy of interval estimates corresponding to variational Bayesian approximations, Proc. 10th. Int. Workshop Artific. Intell. Statist. (Cowell, RG; Ghahramani, Z., eds.) (2005), pp. 373-380

[27] Wang, B.; Titterington, D. M. Variational Bayes Estimation of Mixing Coefficients, Deterministic and statistical methods in machine learning, Volume 3635 (2005), pp. 281-295 | DOI | Zbl

[28] Wang, B.; Titterington, D. M. Convergence properties of a general algorithm for calculating variational Bayesian estimates for a normal mixture model, Bayesian Analysis, Volume 1 (2006) no. 3, pp. 625-650 | DOI | MR | Zbl