Undirected Graphical Models. Raul Queiroz Feitosa
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1 Undirected Graphical Models Raul Queiroz Feitosa
2 Pros and Cons Advantages of UGMs over DGMs UGMs are more natural for some domains (e.g. context-dependent entities) Discriminative UGMs (CRF) are better than discriminative DGMs Advantages of DGMs over UGMs Parameters of DGMs are more interpretable than of UGMs Parameter estimation of UGMs computationally more expensive than of DGMs 6/23/2016 Undirected Graphical Models 2
3 Conditional Independence in UGM Check if A B C A C B Procedure: remove all nodes in C with their links. If there is no path connecting any node in A to any node in B, the conditional independence holds. 6/23/2016 Introduction to Graphical Models 3
4 UGM vs. DGM Do all probability distributions can be perfectly mapped by either UGMs and/or DGMs? DGM UGM all distributions over a given set of variables 6/23/2016 Undirected Graphical Models 4
5 UGM vs. DGM Examples of distributions that can be represented either by an UGM or a DGM: A B A C C A B A B C impossible to represent the same? conditional independence properties D B A B A B C D C D A B 6/23/2016 Undirected Graphical Models 5
6 Clique Clique: any set of fully connected nodes y 1 y 2 y 3 y 4 Maximal clique: a clique such that it is not possible to include any other node from the graph without it ceasing to be a clique. 6/23/2016 Undirected Graphical Models 6
7 Site A site represents a point or a region in the Euclidean space such as an image pixel or an image patch. regular irregular 6/23/2016 Undirected Graphical Models 7
8 Label A countable set of labels x = x 1,, x n, where n is the number of sites, x i 1,, M, M is the number of classes, sky field 6/23/2016 Undirected Graphical Models 8
9 Neighborhood System Examples of neighborhood systems for a regular set of sites S : 4 - Neighborhood 8 - Neighborhood 6/23/2016 Undirected Graphical Models 9
10 Markov Random Field Let x = x i, i S be the labels at an input image, where x i corresponds to the i th site, N i are the set of i neighbors, and S is the set of image sites. Definition: x is said to be a Markov random field on S w.r.t. a neighborhood system N, iff the following two conditions hold for all x i x P x i 0, (positivity) P x i x S i = P x i x Ni, (Markovianity) 6/23/2016 Undirected Graphical Models 10
11 Gibbs Distribution A set of random variables x has a Gibbs distribution on S w.r.t. N iff the lower where P x = 1 Z exp E c x c the higher c C C is the set of all maximal cliques, E c x c >0 is the energy associated with the variables in clique c, Z = x c exp c C E c x c is a normalizing constant called partition function. 6/23/2016 Undirected Graphical Models 11
12 Hammersley-Clifford Theorem x is a Markov random field on S w.r.t. a neighborhood system N, iff P x follows the Gibbs distribution on S w.r.t. a neighborhood system N. 6/23/2016 Undirected Graphical Models 12
13 Gibbs Random Field Example: from the graph below one can write x 5 x 3 x 4 x 1 x 2 P x exp EEx c x c c C E x = E 123 x 1, x 2, x 3 + E 234 x 2, x 3, x 4 +E 35 x 3, x 5 Pairwise MRF: for simplicity it may be restricted to edges rather than to maximal cliques (henceforth): E x = E 12 x 1, x 2 + E 13 x 1, x 3 + E 23 x 2, x 3 + +E 24 x 2, x 4 + E 34 x 3, x 4 + E 35 x 3, x 5 6/23/2016 Undirected Graphical Models 13
14 MAP-MRF framework In the MRF framework, the posterior over the labels x = x i, the corresponding set of observed data y = y i, is expressed using the Bayes rule as, P x y P x p y x where the prior P x over the labels is modeled as a MRF. Usually, for tractability, the likelihood model, p y x is assumed to have the factorized form p y x = p y i x i i S 6/23/2016 Undirected Graphical Models 14
15 Pairwise MRF image model Consequently, the posterior takes the form P x y = 1 Z exp log p y i x i E ij x i, x j i S j N i likelihood given by some generative classifier homogeneous MRF Inference : the classification outcome x is given by x = argmax x P x y 6/23/2016 Undirected Graphical Models 15
16 Example of MRF Potts model: produces a smoothing effect E x i, x j = 0, for x i = x j β, for x i x j (M=3) E = 0 β β β 0 β β β 0 label smoothing Noisy Image Restored Image (ICM) 6/23/2016 Undirected Graphical Models 16
17 Problems with MAP-MRF framework 1) The assumption p y x = i S p y i x i is too restrictive for several natural image analysis applications! 2) The label interaction E x i, x j does not depend on the data y. We may want to turn-off label smoothing between two adjacent nodes if there is a discontinuity in image intensity. 3) To model the posterior P x y we have first to model the likelihood p y i x i. 6/23/2016 Undirected Graphical Models 17
18 Conditional Random Field CRF models the posterior P x y probability directly as an MRF without modeling the prior P x and likelihood p y i x i individually. P x y = 1 Z exp log p x i y II ij x i, x j y i S j N i local posterior given by some discriminative classifier homogeneous CRF 6/23/2016 Undirected Graphical Models 18
19 MRF vs CRF MRF: CRF: P x y = 1 Z exp log p y i x i E x i, x j i S j N i P x y = 1 Z exp log p x i y i I x i, x j y i S j N i 1) Association potentials are likelihoods in MRF and local posteriors in CRF. 2) Association potential is a function of local data in MRF and of all data in CRF. 3) Interaction potential is a function of labels only in MRF and also of data in CRF. 6/23/2016 Undirected Graphical Models 19
20 Example of Interaction Potential Contrast sensitive Potts model: turns off smoothing at data discontinuities, e.g., I x i, x j y i, y j = 0, for x i = x j β 1 D ij maxd ij, for x i x j whereby discontinuity is given by D ij = y i y j 2 6/23/2016 Undirected Graphical Models 20
21 MRF/CRF representation A pairwise MRF/CRF model over a 4 neighborhood regular grid consists of: 1. Node structure (links) Node potentials (unaries) M dimensional vectors Edge potentials (binaries) M M matrices 6/23/2016 Undirected Graphical Models 21
22 MRF/CRF training Unaries are trained in a supervised way Binaries of our examples, i.e., parameter β is estimated by cross-validation. In some cases binaries training also follows a supervised approach. 6/23/2016 Undirected Graphical Models 22
23 Inference It is about computing x = argmax x P x y For a grid containing a b nodes, which may belong to M classes, the number of possibilities is equal to M ab. Exact solution is only feasible for few particular problems. Efficient approximate algorithms are available. To further reduce complexity superpixels instead of pixels are used. 6/23/2016 Undirected Graphical Models 23
24 Practical Assignment Using the Indian Pines hyperspectral dataset, write a MATLAB program that implement a MRF and a CRF model to classify the image. Notice that not all pixels are labeled and training as well test samples are in two separate files. Present the classification results as label images, and report overall and average class accuracies. Hints: For the MRF approach, use the MATLAB function classify to produce the unaries. Notice that not all classifier types will work (why?). For the binaries use the Potts model. For the CRF approach use the MATLAB implementation of Random Forest to produce the unaries. For the binaries use the contrast sensitive Potts model presented in the previous slides. Experiment for some values of β Still regarding the binaries both for the MRF and CRF methods, do not estimate the parameter β using cross validation. It may take to much processing time, though it is the correct approach. Instead, just experiment with some few values of β. For inference both in MRF and CRF use the MATLAB function UGM_Infer_LBP available in UGM library, which must be downloaded to your working folder. The MATLAB function PixelWiseEdgeStruct, will help you to build the Edge Structure. 6/23/2016 Directed Graphical Models 24
25 References Books and Papers Christopher M. Bishop, Patter Recognition and Machine Learning, Springer, Chapter 8. Daphne Koller & Nir Friedman, Probabilistic Graphical Models, MIT Press, Chapter 3. Kevin P. Murphy, Machine Learning A Probabilistic Approach, MIT Press, 2012, Chapter 19. Stan Z. Li, Markov Random Field Modeling in Image Analysis, 3 rd Ed., Springer, Chapter 2. Kumar, S., Hebert, M., Discriminative Random Fields, International Journal of Computer Vision, vol. 68(2), 2006, pp Available at < 6/23/2016 Directed Graphical Models 25
26 References Online lectures Graphical Models 2 Christopher Bishop Graphical Models 3 Christopher Bishop. Probabilistic Graphical Models - Daphne Koller MATLAB Library Schimidt, M., UGM: Matlab code for undirected graphical models, Available at 6/23/2016 Directed Graphical Models 26
27 Undirected Graphical Models END 6/23/2016 Undirected Graphical Models 27
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