Part II. C. M. Bishop PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS
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1 Part II C. M. Bishop PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS
2 Converting Directed to Undirected Graphs (1)
3 Converting Directed to Undirected Graphs (2) Add extra links between all pairs of parents : moralization marrying the parents. The resulting undirected graph, after dropping the arrows, is called the moral graph.
4 Directed vs. Undirected Graphs
5 Inference in Graphical Models A graphical representation of Bayes theorem.
6 Inference on a Chain For N variables, each with K states, there are KN values for x, thus evaluation and storage of the joint distribution, as well as marginalization to obtain p(x n ), all involve storage and computation that scale exponentially with the length N of the chain.
7 Efficient Computation(1) Rearrange the order of the summations and the multiplications allows the required marginal to be evaluated much more efficiently. ab + ac = a(b + c) Doesn t depend on x N
8 Efficient Computation(2) Evaluating the marginal is O(NK 2 ). This is linear in the length of the chain, in contrast to the exponential cost of a naive approach.
9 Inference on a Chain
10 Inference on a Chain
11 Inference on a Chain
12 Inference on a Chain To compute local marginals: Compute and store all forward messages,. Compute and store all backward messages,. Compute Z at any node x m Compute for all variables required.
13 Trees Undirected Tree Directed Tree Polytree
14 Factor Graphs Factors in directed graphs are local conditional distributions. Factors in undirected graphs are potential functions over the maximal cliques (the normalizing coefficient 1/Z can be viewed as a factor defined over the empty set of variables).
15 Factor Graphs from Undirected Graphs Create variable nodes corresponding to the nodes in the original undirected graph Create additional factor nodes corresponding to the maximal cliques x s. The factors f s (x s ) are then set equal to the clique potentials.
16 Factor Graphs from Directed Graphs Create variable nodes in the factor graph corresponding to the nodes of the directed graph. Create factor nodes corresponding to the conditional distributions. Add the appropriate links.
17 The Sum-Product Algorithm (1) Objective: i. to obtain an efficient, exact inference algorithm for finding marginals; ii. in situations where several marginals are required, to allow computations to be shared efficiently. Key idea: Distributive Law
18 The Sum-Product Algorithm (2)
19 The Sum-Product Algorithm (3)
20 The Sum-Product Algorithm (4)
21 The Sum-Product Algorithm (5)
22 The Sum-Product Algorithm (6)
23 The Sum-Product Algorithm (7) Initialization
24 The Sum-Product Algorithm (8) To compute local marginals: Pick an arbitrary node as root Compute and propagate messages from the leaf nodes to the root, storing received messages at every node. Compute and propagate messages from the root to the leaf nodes, storing received messages at every node. Compute the product of received messages at each node for which the marginal is required, and normalize if necessary.
25 Sum-Product: Example (1)
26 Sum-Product: Example (2)
27 Sum-Product: Example (3)
28 Sum-Product: Example (4)
29 The Max-Sum Algorithm (1) Objective: an efficient algorithm for finding i. the value x max that maximises p(x); ii. the value of p(x max ). In general, maximum marginals joint maximum.
30 The Max-Sum Algorithm (2) Maximizing over a chain (max-product)
31 The Max-Sum Algorithm (3) Generalizes to tree-structured factor graph maximizing as close to the leaf nodes as possible
32 The Max-Sum Algorithm (4) Max-Product Max-Sum For numerical reasons, use Again, use distributive law
33 The Max-Sum Algorithm (5) Initialization (leaf nodes) Recursion
34 The Max-Sum Algorithm (6) Termination (root node) Back-track, for all nodes i with l factor nodes to the root (l=0)
35 The Max-Sum Algorithm (7) Example: Markov chain
36 The Junction Tree Algorithm Exact inference on general graphs. Works by turning the initial graph into a junction tree and then running a sumproduct-like algorithm. Intractable on graphs with large cliques.
37 Loopy Belief Propagation Sum-Product on general graphs. Initial unit messages passed across all links, after which messages are passed around until convergence (not guaranteed!). Approximate but tractable for large graphs. Sometime works well, sometimes not at all.
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