D. Figure, Approche du mélange de filtres pour la sélection de mesures dépendant du contexte. L'illustration est donnée dans le cas d'un vecteur d'observation de dimension 2, qui résulte donc en l'exploitation de 3 filtres distincts exploitant respectivement les mesures y 1

. En-pratique, apprentissage ne nécessite ici de définir que les paramètres du réseau de gating. Pour des raisons de simplicité, nous exploitons ici un modèle de région d'activation basé sur des noyaux gaussiens uni-modaux. Ce choix nous permet notamment d'exploiter l'algorithme Expectation Maximisation (EM), 2013.

U. Aspect-important-de-la-méthode-d-'apprentissage-réside-dans-la-nature-structurellement-discriminative-de-ce-modèle, pour optimiser les paramètres du modèle, dans le cas du mélange d'experts comme du mélange de filtres , nous allons chercheràchercherà expliquer au mieux les données d'apprentissage constituées de vecteurs d'observation et de l'´ etat associé. Par conséquent, durant l'apprentissage, nous allons chercheràchercherà expliquer au mieux l'´ etat x t au travers deséquationsdeséquations de filtrage, ce qui n'est pas le cas dans le cadre de l'apprentissage génératif traditionnel, qui cherche luì a optimiser les paramètres du modèlé etat observation demanì erè a expliquer au mieux l'´ evolution de l'´ etatàetatà travers la distribution de prédiction p(x t |x t?1 ) ainsi que les observationsàobservationsà travers la distribution d'observation p(y t |x t ) Notons que cela signifie donc que leséquationsleséquations de filtrage ne sont donc pas exploitées dans le cas de l'apprentissage génératif Discriminative training of kalman filters, Cette spécificité a des conséquences importantes en terme de robustesse, Bibliography Pieter Abbeel Proceedings of Robotics: Science and Systems, 2005.

G. Agamennoni, I. Juan, E. M. Nieto, and . Nebot, An outlierrobust kalman filter, Robotics and Automation (ICRA), 2011 IEEE International Conference on, pp.1551-1558, 2011.

H. James, S. Albert, and . Chib, Bayesian analysis of binary and polychotomous response data, Journal of the American Statistical Association, vol.88, issue.422, pp.669-679, 1993.

A. Allahverdyan and A. Galstyan, Comparative analysis of viterbi training and maximum likelihood estimation for hmms, Advances in Neural Information Processing Systems 24, pp.1674-1682, 2011.

L. Bahl, P. Brown, P. V. De-souza, and R. Mercer, Maximum mutual information estimation of hidden Markov model parameters for speech recognition, ICASSP '86. IEEE International Conference on Acoustics, Speech, and Signal Processing, pp.49-52, 1986.
DOI : 10.1109/ICASSP.1986.1169179

Y. Bar-shalom, T. Kirubarajan, and X. Li, Estimation with Applications to Tracking and Navigation, 2002.
DOI : 10.1002/0471221279

Y. Bengio and P. Frasconi, Input-output HMMs for sequence processing, IEEE Transactions on Neural Networks, vol.7, issue.5, pp.1231-1249, 1996.
DOI : 10.1109/72.536317

URL : http://www.iro.umontreal.ca/~lisa/pointeurs/iohmms.pdf

M. Christopher, J. Bishop, and . Lasserre, Generative or Discrimative? Getting the Best of Both Worlds, Bayesian Statistics 8, pp.3-24, 2007.

C. M. Bishop, Pattern recognition and machine learning, 2006.

H. A. Blom and Y. Bar-shalom, The interacting multiple model algorithm for systems with Markovian switching coefficients, IEEE Transactions on Automatic Control, vol.33, issue.8, pp.780-783, 1988.
DOI : 10.1109/9.1299

X. Boyen and D. Koller, Tractable inference for complex stochastic processes, Proc. UAI, pp.33-42, 1998.

P. Brisset, A. Drouin, M. Gorraz, P. S. Huard, and J. Tyler, The paparazzi solution, Micro Aerial Vehicles, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01004157

K. P. Burnham and D. R. Anderson, Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2002.
DOI : 10.1007/b97636

S. Wassim, R. H. Chaer, J. Bishop, and . Ghosh, A mixture-ofexperts framework for adaptive Kalman filtering, IEEE Transactions on Systems, Man, and Cybernetics, vol.27, 1997.

W. S. Chaer, R. H. Bishop, and J. Ghosh, Hierarchical adaptive kalman filtering for interplanetary orbit determination. Aerospace and Electronic Systems, IEEE Transactions on, vol.34, issue.3, pp.883-896, 1998.
DOI : 10.1109/7.705895

URL : http://www.ideal.ece.utexas.edu/pubs/pdf/1998/chbigh98.pdf

D. Maxwell, C. , and D. Heckerman, Efficient approximations for the marginal likelihood of bayesian networks with hidden variables, Machine Learning, pp.29-181, 1997.

C. Cortes and V. Vapnik, Support-vector networks, Machine Learning, vol.1, issue.3, pp.273-297, 1995.
DOI : 10.1007/BF00994018

D. Crisan and A. Doucet, A survey of convergence results on particle filtering methods for practitioners, IEEE Transactions on Signal Processing, vol.50, issue.3, pp.736-746, 2002.
DOI : 10.1109/78.984773

T. Damoulas, Y. Ying, M. A. Girolami, and C. Campbell, Inferring Sparse Kernel Combinations and Relevance Vectors: An Application to Subcellular Localization of Proteins, 2008 Seventh International Conference on Machine Learning and Applications, pp.577-582, 2008.
DOI : 10.1109/ICMLA.2008.124

T. Damoulas and M. A. Girolami, Probabilistic multi-class multi-kernel learning: on protein fold recognition and remote homology detection, Bioinformatics, vol.1, issue.10, pp.1264-1270, 2008.
DOI : 10.1093/bioinformatics/btl170

URL : https://academic.oup.com/bioinformatics/article-pdf/24/10/1264/16881463/btn112.pdf

T. Damoulas and M. A. Girolami, Combining feature spaces for classification, Pattern Recognition, vol.42, issue.11, pp.2671-2683, 2009.
DOI : 10.1016/j.patcog.2009.04.002

T. Dean and K. Kanazawa, A model for reasoning about persistence and causation, Computational Intelligence, vol.4, issue.2, pp.142-150, 1989.
DOI : 10.1016/0004-3702(87)90012-9

M. Peter-deisenroth, M. F. Huber, and U. D. Hanebeck, Analytic moment-based Gaussian process filtering, Proceedings of the 26th Annual International Conference on Machine Learning, ICML '09, pp.225-232, 2009.
DOI : 10.1145/1553374.1553403

M. Peter-deisenroth, R. D. Turner, M. F. Huber, U. D. Hanebeck, and C. E. Rasmussen, Robust Filtering and Smoothing with Gaussian Processes, IEEE Transactions on Automatic Control, vol.57, issue.7, 1203.
DOI : 10.1109/TAC.2011.2179426

A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum likelihood from incomplete data via the em algorithm, JOURNAL OF THE ROYAL STATISTICAL SOCIETY, SERIES B, vol.39, issue.1, pp.1-38, 1977.

A. Doucet, N. De-freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice, Information Science and Statistics, 2001.
DOI : 10.1007/978-1-4757-3437-9

L. Dovera and E. D. Rossa, Multimodal ensemble Kalman filtering using Gaussian mixture models, Computational Geosciences, vol.12, issue.3, pp.307-323, 2011.
DOI : 10.2118/95750-PA

C. Anita, M. E. Faul, and . Tipping, Analysis of sparse bayesian learning, NIPS, pp.383-389, 2001.

N. Friedman and D. Koller, Being bayesian about network structure: A bayesian approach to structure discovery in bayesian networks, Machine Learning, pp.95-125, 2003.

Z. Ghahramani and G. E. Hinton, Variational Learning for Switching State-Space Models, Neural Computation, vol.3, issue.4, pp.963-996, 1998.
DOI : 10.1162/neco.1997.9.2.227

URL : http://www.gatsby.ucl.ac.uk/Hinton/../publications/papers/11-1999.ps.gz

M. Girolami and S. Rogers, Hierarchic Bayesian models for kernel learning, Proceedings of the 22nd international conference on Machine learning , ICML '05, pp.241-248, 2005.
DOI : 10.1145/1102351.1102382

URL : http://www.dcs.gla.ac.uk/publications/PAPERS/7983/ICML2005_97_final_8_june.pdf

P. W. Goldberg, C. K. Williams, and C. M. Bishop, Regression with input-dependent noise: A gaussian process treatment, Advances in Neural Information Processing Systems 10, pp.493-499

N. J. Gordon and A. F. Smith, Approximate non-gaussian bayesian estimation and modal consistency, Journal of the Royal Statistical Society. Series B (Methodological), vol.55, issue.4, pp.913-918, 1993.

R. A. Jacobs, M. I. Jordan, S. J. Nowlan, and G. E. Hinton, Adaptive Mixtures of Local Experts, Neural Computation, vol.4, issue.1, pp.79-87, 1991.
DOI : 10.1162/neco.1989.1.2.281

A. H. Jazwinski, Stochastic Processes and Filtering Theory, Mathematics in Science and Engineering, 1970.

I. Michael and . Jordan, Hierarchical mixtures of experts and the em algorithm, Neural Computation, vol.6, pp.181-214, 1993.

I. Michael, L. Jordan, and . Xu, Convergence results for the em approach to mixtures of experts architectures, 1993.

S. Biing-hwang-juang and . Katagiri, Discriminative learning for minimum error classification (pattern recognition), IEEE Transactions on Signal Processing, vol.40, issue.12, pp.3043-3054, 1992.
DOI : 10.1109/78.175747

S. Julier and J. K. Uhlmann, A general method for approximating nonlinear transformations of probability distributions, 1996.

J. Simon, J. K. Julier, and . Uhlmann, A new extension of the kalman filter to nonlinear systems, pp.182-193, 1997.

S. J. Julier and J. K. Uhlmann, Unscented Filtering and Nonlinear Estimation, Proceedings of the IEEE, pp.401-422, 2004.
DOI : 10.1109/JPROC.2003.823141

URL : http://www.cs.ubc.ca/~murphyk/Papers/Julier_Uhlmann_mar04.pdf

R. E. Kalman, A New Approach to Linear Filtering and Prediction Problems, Journal of Basic Engineering, vol.82, issue.1, 1960.
DOI : 10.1115/1.3662552

K. Kersting, C. Plagemann, P. Pfaff, and W. Burgard, Most likely heteroscedastic Gaussian process regression, Proceedings of the 24th international conference on Machine learning, ICML '07, pp.393-400, 2007.
DOI : 10.1145/1273496.1273546

URL : http://www.service-robotik-initiative.de/download/publicationen/ALU/kersting07icml.pdf

D. Khashabi, M. Ziyadi, and F. Liang, Heteroscedastic relevance vector machine. CoRR, abs, 1301.

G. Kitagawa, Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, Journal of computational and graphical statistics, vol.5, issue.1, pp.1-25, 1996.
DOI : 10.2307/1390750

G. Kitagawa, Non-Gaussian State-Space Modeling of Nonstationary Time Series, Journal of the American Statistical Association, vol.82, issue.400, pp.1032-1041, 1987.

J. Ko and D. Fox, GP-BayesFilters: Bayesian filtering using Gaussian process prediction and observation models, Autonomous Robots, vol.6, issue.3???4, pp.75-90, 2009.
DOI : 10.1007/s10514-009-9119-x

D. Koller and N. Friedman, Probabilistic Graphical Models: Principles and Techniques -Adaptive Computation and Machine Learning, 2009.

J. Lafferty, Conditional random fields: Probabilistic models for segmenting and labeling sequence data, pp.282-289, 2001.

J. Lafferty, X. Zhu, and Y. Liu, Kernel conditional random fields, Twenty-first international conference on Machine learning , ICML '04, p.64, 2004.
DOI : 10.1145/1015330.1015337

V. Quoc, A. J. Le, and . Smola, Heteroscedastic gaussian process regression, International Conference on Machine Learning ICML, 2005.

J. Loxam and T. Drummond, Student-t Mixture Filter for Robust, Real-Time Visual Tracking, Computer Vision ? ECCV 2008, pp.372-385, 2008.
DOI : 10.1007/11744023_34

URL : http://mi.eng.cam.ac.uk/~jrl39/eccv2008/eccv2008.pdf

M. Lázaro-gredilla and M. K. Titsias, Variational heteroscedastic gaussian process regression, 28th International Conference on Machine Learning (ICML-11, pp.841-848, 2011.

D. J. Mackay, Bayesian Interpolation, Neural Computation, vol.49, issue.3, pp.415-447, 1992.
DOI : 10.1093/comjnl/11.2.185

J. C. David and . Mackay, The evidence framework applied to classification networks, Neural Computation, vol.4, pp.720-736, 1992.

J. C. David and . Mackay, Ensemble learning for hidden markov models, 1997.

P. S. Maybeck, Stochastic Models, Estimation and Control, Mathematics in science and engineering, 1982.

P. S. Maybeck, Stochastic Models, Estimation and Control, Mathematics in science and engineering, 1982.

A. Mccallum, D. Freitag, and F. C. Pereira, Maximum entropy markov models for information extraction and segmentation, Proceedings of the Seventeenth International Conference on Machine Learning, ICML '00, pp.591-598, 2000.

A. Mchutchon and C. E. Rasmussen, Gaussian process training with input noise, NIPS, pp.1341-1349, 2011.

R. K. Mehra, Approaches to adaptive filtering Automatic Control, IEEE Transactions on, vol.17, issue.5, pp.693-698, 1972.
DOI : 10.1109/sap.1970.269992

R. Van, D. Merwe, and E. Wan, Sigma-point kalman filters for probabilistic inference in dynamic state-space models, Proceedings of the Workshop on Advances in Machine Learning, 2003.

P. Thomas and . Minka, Expectation propagation for approximate bayesian inference . CoRR, abs, 1301.

T. Minka, Discriminative models, not discriminative training, 2005.

K. P. Murphy, Machine Learning: A Probabilistic Perspective, p.9780262018029, 2012.

Y. Andrew, M. I. Ng, and . Jordan, On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes, 2001.

J. Pearl, Fusion, propagation, and structuring in belief networks, Artificial Intelligence, vol.29, issue.3, pp.241-288, 1986.
DOI : 10.1016/0004-3702(86)90072-X

J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, 1988.

J. Quiñonero-candela, C. Edward-rasmussen, and R. Herbrich, A unifying view of sparse approximate gaussian process regression, Journal of Machine Learning Research, vol.6, 2005.

V. Ramamurti and J. Ghosh, On the use of localized gating in mixture of experts networks, 1998.

E. Carl, J. Q. Rasmussen, and . Candela, Healing the relevance vector machine through augmentation, Proceedings of the 22nd international conference on Machine learning, ICML '05, pp.689-696, 2005.

A. Ravet, S. Lacroix, and G. Hattenberger, Augmenting Bayes filters with the Relevance Vector Machine for time-varying context-dependent observation distribution, 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp.3039-3044, 2014.
DOI : 10.1109/IROS.2014.6942982

URL : https://hal.archives-ouvertes.fr/hal-01086242

A. Ravet and S. Lacroix, Heterogeneous Bayes Filters with Sparse Bayesian Models: Application to state estimation in robotics, European Conference on Machine Learning and Principles and Practice of Knowledge Discovery (ECML/PKDD), p.10, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01086243

A. Ravet, S. Lacroix, G. Hattenberger, and B. Vandeportaele, Learning to combine multi-sensor information for context dependent state estimation, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp.5221-5226, 2013.
DOI : 10.1109/IROS.2013.6697111

URL : https://hal.archives-ouvertes.fr/hal-00936112

S. Särkkä, Bayesian Filtering and Smoothing, 2013.

S. Sarkka and A. Nummenmaa, Recursive noise adaptive kalman filtering by variational bayesian approximations. Automatic Control, IEEE Transactions on, vol.54, issue.3, pp.596-600, 2009.
DOI : 10.1109/tac.2008.2008348

B. Schölkopf and A. J. Smola, Learning with Kernels: Support Vector Machines , Regularization, Optimization, and Beyond. Adaptive computation and machine learning, 2002.

R. Schubert and G. Wanielik, Unifying bayesian networks and imm filtering for improved multiple model estimation, Information Fusion, 2009. FUSION '09. 12th International Conference on, pp.810-817, 2009.

G. Shafer and J. Pearl, Readings in uncertain reasoning. The Morgan Kaufmann series in representation and reasoning, 1990.

D. Sivia and J. Skilling, Data Analysis: A Bayesian Tutorial, 2006.

C. Sutton and A. Mccallum, An introduction to conditional random fields for relational learning, Introduction to Statistical Relational Learning, 2007.
DOI : 10.1561/2200000013

URL : http://www.cs.umass.edu/%7Ecasutton/publications/crftut-fnt.pdf

A. Thayananthan, Template-based Pose Estimation and Tracking of 3D Hand Motion, 2005.

A. Thayananthan, R. Navaratnam, B. Stenger, P. H. Torr, and R. Cipolla, Multivariate Relevance Vector Machines for Tracking, ECCV (3), pp.124-138, 2006.
DOI : 10.1007/11744078_10

URL : http://mi.eng.cam.ac.uk/~rn246/Publications/paper563_final.pdf

S. Thrun, W. Burgard, and D. Fox, Probabilistic Robotics. Intelligent robotics and autonomous agents, 2005.
DOI : 10.1145/504729.504754

S. Thrun, A probabilistic online mapping algorithm for teams of mobile robots, International Journal of Robotics Research, vol.20, 2001.

S. Thrun, W. Burgard, and D. Fox, Probabilistic Robotics (Intelligent Robotics and Autonomous Agents), 2005.

J. Ting, E. Theodorou, and S. Schaal, Learning an Outlier-Robust Kalman Filter, Machine Learning: ECML 2007, pp.748-756, 2007.
DOI : 10.1007/978-3-540-74958-5_76

URL : http://www-clmc.usc.edu/publications/T/ting-ECML2007.pdf

E. Michael and . Tipping, Sparse bayesian learning and the relevance vector machine, J. Mach. Learn. Res, vol.1, pp.211-244, 2001.

E. Michael, A. Tipping, J. J. Faul, J. Thomson-avenue, and . Avenue, Fast marginal likelihood maximisation for sparse bayesian models, Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, pp.3-6, 2003.

N. Ueda, R. Nakano, Z. Ghahramani, and G. E. Hinton, SMEM Algorithm for Mixture Models, Neural Computation, vol.21, issue.9, pp.2109-2128, 2000.
DOI : 10.1207/s15327906mbr0503_6

URL : http://www.gatsby.ucl.ac.uk/Hinton/../publications/papers/12-1999.ps.gz

A. J. Viterbi, Error bounds for convolutional codes and an asymptotically optimum decoding algorithm, IT, vol.13, issue.2, pp.260-269, 1967.

N. Wiener, Extrapolation, interpolation, and smoothing of stationary time series with engineering applications. Technology press books in science and engineering, 1949.

D. Wipf, J. Palmer, and B. Rao, Perspectives on sparse bayesian learning, Advances in Neural Information Processing Systems, p.2004, 2003.

L. Xu, M. I. Jordan, and G. E. Hinton, An alternative model for mixtures of experts, Advances in Neural Information Processing Systems, 1995.

C. Yuan and C. Neubauer, Variational mixture of gaussian process experts, Advances in Neural Information Processing Systems 21, pp.1897-1904

J. N. Seniha-esen-yuksel, P. D. Wilson, and . Gader, Twenty Years of Mixture of Experts, IEEE Transactions on Neural Networks and Learning Systems, vol.23, issue.8, pp.1177-1193, 2012.
DOI : 10.1109/TNNLS.2012.2200299

J. Shi-cang-zhang, . Li, . Liang-bin, C. Wu, and . Shi, Amultiplemaneuvering targets tracking algorithm based on a generalized pseudo-Bayesian estimator of first order, Journal of Zhejiang University SCIENCE C, vol.14, issue.6, pp.417-424, 2013.
DOI : 10.1631/jzus.C1200310