1. EE462 MLCV Lecture 11-12 Introduction of Graphical Models Markov Random Fields Segmentation Tae-Kyun Kim 1

2. EE462 MLCV Intro: Graphical Models 2 Probabilistic graphical model: is a diagrammatic representation of probability distributions Advantages: 1.It provides a simple way to visualise the structure of a probabilistic model and can be used to design and motivate new models. 2.Insights into the properties of the model, including conditional independence properties, can be obtained. 3.Complex computations of inference and learning can be expressed in terms of graphical manipulations.

3. EE462 MLCV 3 A graph comprises nodes (or vertices) and links (or edges), where each node represents a random variable and the link probabilistic relationship between these variables. The graph captures the joint distribution of all random variables. Bayesian networks: are directed graphical models. Markov random fields: are as undirected graphical models. Inference (exact or approximate) is the process of computing posterior probabilities of one or more nodes.

4. EE462 MLCV 4 Bayesian Networks Consider three variables a,b,c. The joint distribution can be written as This can be represented by a graphical model as The node a is called the parent node of b, and b is the child of the node a.

5. EE462 MLCV 5 … Consider K variables whose joint distribution is given The graphical model of this formulation has a link between every pair of nodes, and is thus fully connected.

6. EE462 MLCV 6 See below for the case of absence of some links. The joint distribution is given as

7. EE462 MLCV 7 The joint distribution for a graph with K nodes can be written in the form of where pa k denotes the set of parents of x k, and x = {x 1,...,x K }. The equation expresses the factorization properties of the joint distribution for a directed graphical model.

8. EE462 MLCV Examples: Bayesian Polynomial Regression (discriminative) Gaussian Mixture Models (generative) 8

9. EE462 MLCV Polynomial curve fitting (recap)

10. EE462 MLCV 10 The Bayesian polynomial regression model is represented by the directed graphical model as A more compact representation for the same is by using a plate (the box labelled N) that represents N nodes of t n. Bayesian polynomial regression Lecture 15-16

11. EE462 MLCV 11 The nodes {t n } (and also x n ) are shaded to indicate that the corresponding random variables are set to their observed (training set) values. Bayesian polynomial regression

12. EE462 MLCV 12 See below for the graphical model of GMM. Generative (unsupervised) vs Discriminative (supervised) models: In GMM, the input variable x n receives arrows and is thus generative. The regression model maps the input x n to the target t n, it thus takes an arrow in the direction from x n to t n (discriminative). Gaussian Mixture Model (GMM)

13. EE462 MLCV 13 Conditional Independence The variable a is conditionally independent of b given c, if or The variables a,b are statistically independent given c. They are notated as

14. EE462 MLCV 14 See below for the example. The joint distribution in the graph is given as If none of the variables are observed, and, in general, is not factorized into p(a)p(b), and so

15. EE462 MLCV 15 Suppose we condition on the variable c, we can write down

16. EE462 MLCV 16 The example of Markov chain is given below. The joint distribution is given as thus, is not factorized into p(a)p(b), and so

17. EE462 MLCV 17 So, we get the conditional independence property Suppose we condition on the variable c, we can write down

18. EE462 MLCV This will help graph separation or factorization, then inference. 18

19. EE462 MLCV 19 Markov Random Fields An example of an undirected graph in which every path from any nodes in set A to any node in set B passes through at least one node in set C. Consequently the conditional independence property A ⊥ B|C holds.

20. EE462 MLCV 20 A clique is a subset of the nodes in a graph such that there exists a link between all pairs of nodes in the subset. A maximal clique is a clique such that it is not possible to include any other nodes without it ceasing to be a clique. The two maximal cliques in the example are

21. EE462 MLCV 21 Let us denote a maximal clique by C and the set of variables in the clique by x C. The joint distribution is given as a product of potential functions Ψ C (x C ) over the maximal cliques of the graph as where Z is a normalisation constant. We do not restrict the choice of potential functions to those of marginal or conditional probability distributions in contrast to directed graphs. Markov Random Fields

22. EE462 MLCV 22 where E(x C ) is called an energy function. Markov Random Fields We define strictly positive potential functions Ψ C (x C ), e.g.

23. EE462 MLCV 23 Markov Random Fields for Image De-noising

24. EE462 MLCV 24 Markov Random Fields for Image De-noising The original image is corrupted by flipping the sign of the pixels with probability 10%. Our goal is to recover the original noise-free image.

25. EE462 MLCV 25 The undirected graphical model of MRF for image de- noising: x i is a binary variable denoting the state of pixel i in the unknown noise-free image. y i denotes the corresponding value of pixel i in the observed noisy image.

26. EE462 MLCV 26 The energy function to minimise for the model is where the first term is a bias (or prior) term. The joint distribution over x and y are given by Markov Random Fields for Image De-noising *The above example was obtained by β = 1.0, η = 2.1 and h = 0. It means the equal prior probabilities for the two states of x i.

27. EE462 MLCV 27 It initialises the variables x i by x i = y i for all i. For all i, it takes one node x i at a time and evaluates the total energy for the two possible states x i = +1 and x i = -1, keeping all other node variables fixed. it sets x i to whichever state has the lower energy. The solution: Iterated conditional models (ICM) It iterates the following until convergence.

28. EE462 MLCV Image De-Noising Demo http://homepages.inf.ed.ac.uk/rbf/CVon line/LOCAL_COPIES/AV0809/ORCHA RD/ 28