Prof. Adriana Kovashka University of Pittsburgh April 4, 2017

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Presentation transcript:

Prof. Adriana Kovashka University of Pittsburgh April 4, 2017 CS 2750: Machine Learning Markov Random Fields Inference in Graphical Models Prof. Adriana Kovashka University of Pittsburgh April 4, 2017

Bayes Nets vs. Markov Nets Bayes nets represent a subclass of joint distributions that capture non-cyclic causal dependencies between variables. A Markov net can represent any joint distribution. Ray Mooney

Markov Random Fields Undirected graph over a set of random variables, where an edge represents a dependency. The Markov blanket of a node, X, in a Markov Net is the set of its neighbors in the graph (nodes that have an edge connecting to X). Every node in a Markov Net is conditionally independent of every other node given its Markov blanket. Ray Mooney

Markov Random Fields Markov Blanket A node is conditionally independent of all other nodes conditioned only on the neighboring nodes. Chris Bishop

Cliques and Maximal Cliques Chris Bishop

Joint Distribution for a Markov Net The distribution of a Markov net is most compactly described in terms of a set of potential functions, ψc, for each clique, C, in the graph. For each joint assignment of values to the variables in clique C, ψc assigns a non-negative real value that represents the compatibility of these values. Ray Mooney

Joint Distribution for a Markov Net where is the potential over clique C and is the normalization coefficient; note: M K-state variables  KM terms in Z. Energies and the Boltzmann distribution Chris Bishop

Illustration: Image De-Noising Original Image Noisy Image Chris Bishop

Illustration: Image De-Noising yi in {+1, -1}: labels in noisy image (which we have), xi in {+1, -1}: labels in noise-free image (which we want to recover), i is the index over pixels xj Prior Pixels are like their neighbors Pixels of noisy and noise-free images are related Adapted from Chris Bishop

Illustration: Image De-Noising Noisy Image Restored Image (ICM) Chris Bishop

Inference on a Chain O(KN) operations (K states, N variables) Adapted from Chris Bishop

Inference on a Chain O(NK2) operations (K states, N variables) Chris Bishop

Inference on a Chain Chris Bishop

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 xm Compute for all variables required. Chris Bishop

Factor Graphs Chris Bishop

Factor Graphs from Directed Graphs Chris Bishop

Factor Graphs from Undirected Graphs Chris Bishop

The Sum-Product Algorithm Objective: to obtain an efficient, exact inference algorithm for finding marginals; in situations where several marginals are required, to allow computations to be shared efficiently. Key idea: Distributive Law Chris Bishop

The Sum-Product Algorithm 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. Chris Bishop

Sum-Product: Example Chris Bishop

Sum-Product: Example fa fb fc Chris Bishop

Sum-Product: Example fa fb fc Chris Bishop

Sum-Product: Example Chris Bishop

The Max-Sum Algorithm Objective: an efficient algorithm for finding the value xmax that maximises p(x); the value of p(xmax). In general, maximum marginals  joint maximum. Chris Bishop