Introduction to probabilistic graphical models

Class projects


The project gives you the opportunity to study in greater depth certain concepts of the course. The topic has to be linked with algorithms, concepts or methods presented in class, but beyond this requirement, the choice is quite open. In particular, it may be tailored to your interests.

The standard class projects need to contain the following 3 components:

  1. An article review around a given topic (research articles or "polycopie" not studied in class). See below for a list of tentative projects. This means to read and understand a specific research article.
  2. An implementation of the method.
  3. An experimentation with real data. This means to apply the method on real data and report your findings and observations. If the paper is quite dense and theoretical, then an experimentation on simulated / synthetic data is sufficient.

The project should be done in groups of three students (to the extend possible). Groups of four are also exceptionally acceptable, but then we expect a more involved project justifying the number of people. For students who are asking for recommendation letters, please ask us about specific project requirements. Once you have an idea of a project, it is mandatory to have it validated by the teachers (by email, see instructions below) because we would like the projects undertaken to be as diverse as possible.

Note: the project may be done jointly with another class (with the formal agreement of the corresponding teachers). For joint projects between GM and Object recognition and computer vision course, get in touch with I. Laptev and J. Sivic. Note that joint projects have to be more ambitious and have to meet the expectations of both courses. So they are recommended only if you are very motivated.


The final class project makes a third of the final grade for the class. Evaluation will be made on:
    * A report (less than 6 pages) presenting the project and the obtained results (for applicative projects), to be given by January 11th, 2017. The report has to be written in such a way that any student who has followed the class can understand (no need to introduce graphical model concepts). The report has to clearly present (in French or English) the studied problem and the existing approaches. You will be more evaluated on the clarity of the report rather than on its length.
    * A poster presentation of 5 minutes, from at most 8 A4 pages which will displayed on the wall (Wednesday January 4th, 2017, from 9am to 12pm in room C102-C103, Batiment Cournot, ENS Cachan). The poster should be in English. The presentation (in French or English) is also geared towards other students and the goal is to highlight in the allocated time the salient points. Like in a regular conference poster session, students are encouraged to attend other student posters.

(1) Each student or group of students has to obtained the agreement of teachers (by email) before November 18th, 2016.
(2) Each student or group of students has to submit a small mid-project report (one page, as a latex or text document) presenting the progress and the obtained results, so that some feedback may be given. This report has to submitted on the Moodle before December 9th, 2016.


The various steps are summarized below.

November Choose a project (one or two students per projects, preferably two)
Before 11/18 Send an email to the three teachers, in which all members of your team are cc'ed to request our agreement on your choice of team and project topic.
Before 12/09 Send a draft (1 page) + first results, on the Moodle.
On 2017/01/04 Poster session in Batiment Cournot (C102-103) - 9am to 12pm
Before 2017/01/11 Submit your project report (~6 pages, on the Moodle)

Article or chapter review

The goal of this list of projects is to show classical (and if possible interesting) articles using or improving graphical models. This may give you an idea of current research topics as well as applicative projects. Even if you don't select any of these, reading some of them is advised.

Probabilistic PCA
Probabilistic CCA
Interpretation of PCA as a graphical model close to factorial analysis. A situation where EM has no local minima.

Tipping, M. E.,  Bishop, C. M. 1999. Probabilistic principal component analysis. Journal of the Royal Statistical Society, Series B 61(3):611-622. [pdf]

CCA (Canonical Correlation Analysis) is analogous to PCA for the joint analysis of two random vectors X and Y.
  1. D. R Hardoon, S. Szedmak, et J. Shawe-Taylor, Canonical correlation analysis: an overview with application to learning methodsNeural Computation 16, no. 12 (2004): 2639-2664. 
  2. F. R Bach et M. I Jordan, A probabilistic interpretation of canonical correlation analysis,Dept. Statist., Univ. California, Berkeley, CA, Tech. Rep 688 (2005). 
Learning graph structure - multinomial models
For complete discrete data, learning of parameters and directed acyclic graph.

D. Heckerman, D. Geiger, D. Chickering. Learning Bayesian networks: The Combination of Knowledge and Statistical Data.  Machine Learning, 20:197-243, 1995.

Learning graph structure - Gaussian models
For complete Gaussian data, learning of parameters and directed acyclic graph.

D. Geiger, D. Heckerman. Learning Gaussian networks.  Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence, pp. 235--243.
Variational methods for inference Class of method for approximate inference.

An introduction to variational methods for graphical models. M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. In M. I. Jordan (Ed.), Learning in Graphical Models, Cambridge: MIT Press, 1999

Its application to Bayesian inference.

Beal, M.J. and Ghahramani, Z.
Variational Bayesian Learning of Directed Graphical Models with Hidden Variables
To appear in Bayesian Analysis 1(4), 2006.
Simulation methods for inference (particle filtering)
A simulation for dynamic graphical models
Chapter of "polycopie"

S. Arulampalam, S. Maskell, N. J. Gordon, and T. Clapp, A Tutorial on Particle Filters for On-line Non-linear/Non-Gaussian Bayesian Tracking, IEEE Transactions of Signal Processing, Vol. 50(2), pages 174-188, February 2002.

Doucet A., Godsill S.J. and Andrieu C., "On sequential Monte Carlo sampling methods for Bayesian filtering," Statist. Comput., 10, 197-208, 2000
Semi-Markovian models
A class of models allowing to model the time spent in any given state for a Markov Chain and an HMM.

Note from Kevin Murphy [pdf]

Learning parameters in an undirected graphical model (Markov random fields) Chapter 9 of "polycopie" and articles.

Dynamic graphical models

Chapter of "polycopie". Specific topics to be defined.
General applications of the sum-product algorithms (e.g., to the FFT) The generalized distributive law, S. M. Aji, R. J. Mceliece
Information Theory, IEEE Transactions on, Vol. 46, No. 2. (2000), pp. 325-343.
Independent Component Analysis A. Hyvarinen, E. Oja (2000): Independent Component Analysis: Algorithms and Application, Neural Networks, 13(4-5):411-430, 2000.

Course of Herve LeBorgne:
Jean-Francois Cardoso Dependence, Correlation and Gaussianity in Independent Component Analysis. Journal of Machine Learning Research 4(Dec):1177--1203, 2003. [pdf]

Clustering through a mixture of PCA

M. E Tipping et C. M Bishop, Mixtures of probabilistic principal component analyzers, Neural computation 11, no. 2 (1999): 443-482.

Stochastic relational models
  • E. M Airoldi et al., Mixed membership stochastic block models for relational data with application to protein-protein interactions, dans Proceedings of the International Biometrics Society Annual Meeting, 2006.

  • Conditional Random Fields
    Charles Sutton, Andrew McCallum An Introduction to Conditional Random Fields for Relational Learning . In Lise Getoor and Ben Taskar, editors, Introduction to Statistical Relational Learning. MIT Press. 2007.

    Dirichlet Process
  • R. M Neal, Markov chain sampling methods for Dirichlet process mixture models, Journal of computational and graphical statistics 9, no. 2 (2000): 249-265. 
  • Y. W Teh,  Dirichlet Process,  Submitted to Encyclopedia of Machine Learning (2007). 
  • A. Ranganathan,  The Dirichlet Process Mixture (DPM) Model (2004).

  • Factorial HMM

    Z. Ghahramani et M. I Jordan,  Factorial hidden Markov modelsMachine learning 29, no. 2 (1997): 245-273

    Generalized PCA
    M. Collins, S. Dasgupta, et R. E Schapire, A generalization of principal component analysis to the exponential family, Advances in neural information processing systems 1 (2002): 617-624. 

    Structure learning by L1 regularization

    J. Friedman, T. Hastie, et R. Tibshirani, Sparse inverse covariance estimation with the graphical lasso, Biostatistics 9, no. 3 (2008): 432. 

    Mixture of log-concave densities

     Interesting non-parametric models for unimodal distributions.

  • T Chang et G. Walther,  Clustering with mixtures of log-concave
    , Computational Statistics & Data Analysis 51, no. 12
    (2007): 6242-6251.
  • G. Walther, Inference and modeling with log-concave distributions, Statistical Science 25 (2010).

  • Learning non-linear
    dynamical systems

     Idea: use non-linear Kalman filters to learn the dynamics of a non-linear dynamical system form noisy observation of the system:
    Roweis, Sam, and Zoubin Ghahramani. Learning Nonlinear Dynamical Systems Using the Expectation–maximization Algorithm. Kalman Filtering and Neural Networks 6 (2001): 175–220.

    Hawkes process estimation
     Hawkes processes are stochastic processes that allow to describe cascade or networks of events that trigger one another via delayed self-exitation. It is possible to design an EM algorithm to learn the structure of the stochastic process
    Lewis, E., & Mohler, G. (2011). A nonparametric EM algorithm for multiscale Hawkes processes. Preprint.

    Particle Filtering
    Christian A. Naesseth, Fredrik Lindsten, Thomas B. Schön (2014),
    Sequential Monte Carlo for Graphical Models

    See also the section on particle filter in chapter 21 of the polycopié

    Examples of applications of Graphical models

    These articles present classical applications. They may give you ideas for an applicative project or may be used for article reviews.

    Bioinformatics Chapter of polycopie (ask login/pwd by e-mail)

    Phylogenetic HMM:
    A. Siepel et D. Haussler, Phylogenetic hidden Markov models, Statistical methods in molecular evolution (2005), 3, 325-351. 

    Vision/Speech Articles from Kevin Murphy:
    "Using the Forest to See the Trees:A Graphical Model Relating Features, Objects and Scenes"  Kevin Murphy, Antonio Torralba, William Freeman. NIPS'03 (Neural Info. Processing Systems)

    Dynamic Bayesian Networks for Audio-Visual Speech Recognition  A. Nefian, L. Liang, X. Pi, X. Liu and K. Murphy. EURASIP, Journal of Applied Signal Processing, 11:1-15, 2002

    Optimization for MAP inference in computer vision:
    MRF Optimization via Dual Decomposition: Message-Passing Revisited, Komodakis, Paragios, Tziritas, ICVV 2007. Longer technical report version

    Learning tree structured context model for object recognition:

    In object recognition applying independent single-object detectors may produce semantically incorrect results and even perfect detectors cannot solve context-related tasks in scene understanding. With a probabilistic graphical model it is possible to capture contextual information of a scene and apply it to object recognition and scene understanding problems.
    Choi, M. J., Torralba, A., & Willsky, A. S. (2012). A tree-based context model for object recognition. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 34(2), 240-252.

    Robotics Automatic construction of maps
    Simultaneous Localization and Mapping with Sparse Extended Information Filters
    Thrun et al. The International Journal of Robotics Research.2004; 23: 693-716.
    (see also chapter of "polycopie" on Kalman filtering)
    Text Naive Bayes:
    A. McCallum and K. Nigam. A comparison of event models for Naive Bayes text classification. In AAAI-98 Workshop on Learning for Text Categorization, 1998. 

    Latent Dirichlet allocation. D. Blei, A. Ng, and M. Jordan. Journal of Machine Learning Research, 3:993-1022, January 2003. [ .ps.gz | .pdf | code ]
    Text - Natural language processing S. Vogel, H. Ney, and C. Tillmann. HMM-based word alignment in statistical translation. In Proceedings of the 16th conference on Computational linguistics, pp. 836-841, Morristown, NJ, USA, 1996. Association for Computational Linguistics.

    Non contextual probabilistic grammars:
    Notes de cours de CMU, 1999

    Deep learning and graphical models

    Connectionnist Temporal Classification

    An algorithm for recurrent neural network for classification along a sequence:
    Graves, A., Fernández, S., Gomez, F., & Schmidhuber, J. (2006).
    Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks. In Proceedings of the 23rd International Conference on Machine Learning (pp. 369-376). ACM.

    Restricted Boltzman Machines

    A probabilistic form of neural network:

    Fischer, A., & Igel, C. (2014). Training restricted Boltzmann machines: an introduction. Pattern Recognition, 47(1), 25-39.

    Implementation of algorithms

    N most probable configurations Implementation of an algorithm (HMM or more complex graphs), from the following articles:
    Dennis Nilsson, Jacob Goldberger
    An Efficient Algorithm for Sequentially finding the N-Best List , IJCAI, 1999
    Chen Yanover, Yair Weiss, Finding the M Most Probable Configurations Using Loopy Belief Propagation, NIPS 2003.
    Computation of tree-width Comparing the classical heuristics and finer methods:

    Mark Hopkins and Adnan Darwiche
    A Practical Relaxation of Constant-Factor Treewidth Approximation Algorithms
    Proceedings of the First European Workshop on Probabilistic Graphical Models 2002

    Also some exact methods
    Stefan Arnborg, Derek G. Corneil, Andrzej Proskurowski, Complexity of finding embeddings in a k-tree, SIAM Journal on Algebraic and Discrete Methods (1997)