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Pattern Recognition and Machine Learning : Graphical Models

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Pattern Recognition and Machine Learning Chapter 8: graphical models
Bayesian Networks Directed Acyclic Graph (DAG)
Bayesian Networks General Factorization
Bayesian Curve Fitting (1) Polynomial
Bayesian Curve Fitting (2) Plate
Bayesian Curve Fitting (3) Input variables and explicit hyperparameters
Bayesian Curve Fitting —Learning Condition on data
Bayesian Curve Fitting —Prediction Predictive distribution: where
Generative Models Causal process for generating images
Discrete Variables (1) General joint distribution: K 2 { 1 parameters Independent joint distribution: 2(K{ 1) parameters
Discrete Variables (2) General joint distribution over M variables: KM{ 1 parameters M -node Markov chain: K{ 1 + (M{ 1) K(K{ 1) parameters
Discrete Variables: Bayesian Parameters (1)
Discrete Variables: Bayesian Parameters (2) Shared prior
Parameterized Conditional Distributions If are discrete, K-state variables, in general has O(K M) parameters. The parameterized form requires only M+ 1 parameters
Linear-Gaussian Models Directed Graph Vector-valued Gaussian Nodes Each node is Gaussian, the mean is a linear function of the parents.
Conditional Independence a is independent of b given c Equivalently Notation
Conditional Independence: Example 1
Conditional Independence: Example 1
Conditional Independence: Example 2
Conditional Independence: Example 2
Conditional Independence: Example 3 Note: this is the opposite of Example 1, with c unobserved.
Conditional Independence: Example 3 Note: this is the opposite of Example 1, with c observed.
“Am I out of fuel?” B = Battery (0=flat, 1=fully charged) F = Fuel Tank (0=empty, 1=full) G = Fuel Gauge Reading (0=empty, 1=full) and hence
“Am I out of fuel?” Probability of an empty tank increased by observing G = 0.
“Am I out of fuel?” Probability of an empty tank reduced by observing B = 0. This referred to as “explaining away”.
D-separation ,[object Object]
A path from A to B is blocked if it contains a node such that eitherthe arrows on the path meet either head-to-tail or tail-to-tail at the node, and the node is in the set C, or the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C. ,[object Object]
If A is d-separated from B by C, the joint distribution over all variables in the graph satisfies .,[object Object]
D-separation: I.I.D. Data
Directed Graphs as Distribution Filters
The Markov Blanket Factors independent of xi cancel between numerator and denominator.
Markov Random Fields Markov Blanket
Cliques and Maximal Cliques Clique Maximal Clique
Joint Distribution where is the potential over clique C and is the normalization coefficient; note: MK-state variables  KM terms in Z. Energies and the Boltzmann distribution
Illustration: Image De-Noising (1) Original Image Noisy Image
Illustration: Image De-Noising (2)
Illustration: Image De-Noising (3) Noisy Image Restored Image (ICM)
Illustration: Image De-Noising (4) Restored Image (Graph cuts) Restored Image (ICM)
Converting Directed to Undirected Graphs (1)
Converting Directed to Undirected Graphs (2) Additional links
Directed vs. Undirected Graphs (1)
Directed vs. Undirected Graphs (2)
Inference in Graphical Models
Inference on a Chain
Inference on a Chain
Inference on a Chain
Inference on a Chain
Inference on a Chain To compute local marginals: ,[object Object]
Compute and store all backward messages, .
Compute Z at any node xm
Computefor all variables required.,[object Object]
Factor Graphs
Factor Graphs from Directed Graphs
Factor Graphs from Undirected Graphs
The Sum-Product Algorithm (1) 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
The Sum-Product Algorithm (2)
The Sum-Product Algorithm (3)
The Sum-Product Algorithm (4)
The Sum-Product Algorithm (5)
The Sum-Product Algorithm (6)
The Sum-Product Algorithm (7) Initialization
The Sum-Product Algorithm (8) To compute local marginals: ,[object Object]
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.,[object Object]
Sum-Product: Example (2)
Sum-Product: Example (3)
Sum-Product: Example (4)
The Max-Sum Algorithm (1) Objective: an efficient algorithm for finding the value xmax that maximises p(x); the value of p(xmax). In general, maximum marginals  joint maximum.
The Max-Sum Algorithm (2) Maximizing over a chain (max-product)
The Max-Sum Algorithm (3) Generalizes to tree-structured factor graph maximizing as close to the leaf nodes as possible

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