Probabilistic Graphical Models

Size: px
Start display at page:

Download "Probabilistic Graphical Models"

Transcription

1 Probabilistic Graphical Models Brown University CSCI 295-P, Spring 213 Prof. Erik Sudderth Lecture 11: Inference & Learning Overview, Gaussian Graphical Models Some figures courtesy Michael Jordan s draft textbook, An Introduction to Probabilistic Graphical Models

2 Graphical Models, Inference, Learning Graphical Model: A factorized probability representation Directed: Sequential, causal structure for generative process Undirected: Associate features with edges, cliques, or factors Inference: Given model, find marginals of hidden variables Standardize: Convert directed to equivalent undirected form Sum-product BP: Exact for any tree-structured graph Junction tree: Convert loopy graph to consistent clique tree x 1 x 2 x 3 x 4 x 5

3 Undirected Inference Algorithms One Marginal All Marginals Tree elimination applied recursively to leaves of tree belief propagation or sum-product algorithm Graph elimination algorithm junction tree algorithm: belief propagation on a junction tree A junction tree is a clique tree with special properties: Ø Consistency: Clique nodes corresponding to any variable from the original model form a connected subtree Ø Construction: Triangulations and elimination orderings

4 Graphical Models, Inference, Learning Graphical Model: A factorized probability representation Directed: Sequential, causal structure for generative process Undirected: Associate features with edges, cliques, or factors Inference: Given model, find marginals of hidden variables Standardize: Convert directed to equivalent undirected form Sum-product BP: Exact for any tree-structured graph Junction tree: Convert loopy graph to consistent clique tree Learning: Given a set of complete observations of all variables Directed: Decomposes to independent learning problems: Predict the distribution of each child given its parents Undirected: Global normalization globally couples parameters: Gradients computable by inferring clique/factor marginals Learning: Given a set of partial observations of some variables E-Step: Infer marginal distributions of hidden variables M-Step: Optimize parameters to match E-step and data stats

5 Learning for Undirected Models Undirected graph encodes dependencies within a single training example: p(d ) = NY n=1 1 Z( ) p(x ) =exp X f2f f (x f f )=exp{ T f f (x f )} T f f (x f ) A( ) D = {x V,1,...,x V,N } Given N independent, identically distributed, completely observed samples: log p(d ) = Y f2f " N X n=1 f (x f,n f ) X f2f T f f (x f,n ) # NA( ) A( ) = log Z( )

6 Learning for Undirected Models Undirected graph encodes dependencies within a single training example: p(d ) = NY n=1 1 Z( ) Y f2f f (x f,n f ) D = {x V,1,...,x V,N } Given N independent, identically distributed, completely observed samples: log p(d ) = Take gradient with respect to parameters for a single factor: r f log p(d ) = " N X n=1 " N X n=1 X f2f T f f (x f,n ) f (x f,n ) # NE [ f (x f )] Must be able to compute marginal distributions for factors in current model: Ø Tractable for tree-structured factor graphs via sum-product Ø For general graphs, use the junction tree algorithm to compute # NA( )

7 Undirected Optimization Strategies log p(d ) = r f log p(d ) = " N X n=1 " X N n=1 X f2f T f f (x f,n ) f (x f,n ) # # NA( ) NE [ f (x f )] Gradient Ascent: Quasi-Newton methods like PCG, L-BGFS, Gradients: Difference between statistics of observed data, and inferred statistics for the model at the current iteration Objective: Explicitly compute log-normalization (variant of BP) Coordinate Ascent: Maximize objective with respect to the parameters of a single factor, keeping all other factors fixed Simple closed form depending on ratio between factor marginal for current model, and empirical marginal from data Iterative proportional fitting (IPF) and generalized iterative scaling algorithms (t+1) f (x f )= (t) f (x f ) p(x f ) p (t) f (x f )

8 Advanced Topics on the Horizon Graph Structure Learning f (x f f )=exp{ T f f (x f )} Setting factor parameters to zero implicitly removes from model Feature selection: Search-based, sparsity-inducing priors, Topologies: Tree-structured, directed, bounded treewidth, Approximate Inference: What if junction tree is intractable? Simulation-based (Monte Carlo) approximations Optimization-based (variational) approximations Inner loop of algorithms for approximate learning Alternative Objectives Max-Product: Global MAP configuration of hidden variables Discriminative learning: CRF, max-margin Markov network, Inference with Continuous Variables Gaussian: Closed form mean and covariance recursions Non-Gaussian: Variational and Monte Carlo approximations

9 Pairwise Markov Random Fields p(x) = 1 Z Y (s,t)2e st(x s,x t ) Y s2v s(x s ) x 1 x 2 Simple parameterization, but still expressive and widely used in practice Guaranteed Markov with respect to graph Any jointly Gaussian distribution can be represented by only pairwise potentials x 3 x 4 x 5 E V Z set of undirected edges (s,t) linking pairs of nodes set of N nodes or vertices, {1, 2,...,N} normalization constant (partition function)

10 Inference in Undirected Trees For a tree, the maximal cliques are always pairs of nodes: p(x) = 1 Z Y (s,t)2e st(x s,x t ) Y s(x s ) s2v

11 Belief Propagation (Integral-Product) BELIEFS: Posterior marginals x t MESSAGES: Sufficient statistics m ts (x s ) / Z ˆp t (x t ) / t (x t ) Y (t) x t st(x s,x t ) t (x t ) u2 (t) m ut (x t ) neighborhood of node t (adjacent nodes) Y u2 (t)\s m ut (x t ) x s x t I) Message Product II) Message Propagation

12 BP for Continuous Variables Is there a finitely parameterized, closed form for the message and marginal functions? Is there an analytic formula for the message integral, phrased as an update of these parameters?

13 Covariance and Correlation Covariance: 2 R d d u T u for any u 2 R d 1,u6= Always positive semidefinite: Often positive definite: Correlation: u T u> for any u 2 R d 1,u6= Independence:

14 Gaussian Distributions Simplest joint distribution that can capture arbitrary mean & covariance Justifications from central limit theorem and maximum entropy criterion Probability density above assumes covariance is positive definite ML parameter estimates are sample mean & sample covariance

15 Two-Dimensional Gaussians 5 spherical 1 diagonal 6 full full spherical diagonal

16 Gaussian Geometry Eigenvalues and eigenvectors: u i = i u i,i=1,...,d For a symmetric matrix: i 2 R u T i u i =1 u T i u j = dx =U U T = iu i u T i i=1 For a positive semidefinite matrix: i For a positive definite matrix: i > 1 = U 1 U T = dx 1 i=1 i u i u T i y i = u T i (x µ)

17 Probabilistic PCA & Factor Analysis Both Models: Data is a linear function of low-dimensional latent coordinates, plus Gaussian noise p(x i z i, )=N (x i Wz i + µ, ) p(x i ) =N (x i µ, W W T + ) Factor analysis: Probabilistic PCA: x 2 is a general diagonal matrix is a multiple of identity matrix = 2 I p(x bz) w p(z i ) =N (z i,i) x 2 low rank covariance parameterization µ } b z w µ p(z) p(x) bz z x 1 C. Bishop, Pattern Recognition & Machine Learning x 1

18 Gaussian Graphical Models x N (µ, ) J = 1

19 Gaussian Potentials

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Brown University CSCI 2950-P, Spring 2013 Prof. Erik Sudderth Lecture 12: Gaussian Belief Propagation, State Space Models and Kalman Filters Guest Kalman Filter Lecture by

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Brown University CSCI 2950-P, Spring 2013 Prof. Erik Sudderth Lecture 13: Learning in Gaussian Graphical Models, Non-Gaussian Inference, Monte Carlo Methods Some figures

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Brown University CSCI 2950-P, Spring 2013 Prof. Erik Sudderth Lecture 9: Expectation Maximiation (EM) Algorithm, Learning in Undirected Graphical Models Some figures courtesy

More information

14 : Theory of Variational Inference: Inner and Outer Approximation

14 : Theory of Variational Inference: Inner and Outer Approximation 10-708: Probabilistic Graphical Models 10-708, Spring 2017 14 : Theory of Variational Inference: Inner and Outer Approximation Lecturer: Eric P. Xing Scribes: Maria Ryskina, Yen-Chia Hsu 1 Introduction

More information

Introduction to Machine Learning

Introduction to Machine Learning Introduction to Machine Learning Brown University CSCI 1950-F, Spring 2012 Prof. Erik Sudderth Lecture 25: Markov Chain Monte Carlo (MCMC) Course Review and Advanced Topics Many figures courtesy Kevin

More information

STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 11 Project

More information

p L yi z n m x N n xi

p L yi z n m x N n xi y i z n x n N x i Overview Directed and undirected graphs Conditional independence Exact inference Latent variables and EM Variational inference Books statistical perspective Graphical Models, S. Lauritzen

More information

6.867 Machine learning, lecture 23 (Jaakkola)

6.867 Machine learning, lecture 23 (Jaakkola) Lecture topics: Markov Random Fields Probabilistic inference Markov Random Fields We will briefly go over undirected graphical models or Markov Random Fields (MRFs) as they will be needed in the context

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Lecture 11 CRFs, Exponential Family CS/CNS/EE 155 Andreas Krause Announcements Homework 2 due today Project milestones due next Monday (Nov 9) About half the work should

More information

13: Variational inference II

13: Variational inference II 10-708: Probabilistic Graphical Models, Spring 2015 13: Variational inference II Lecturer: Eric P. Xing Scribes: Ronghuo Zheng, Zhiting Hu, Yuntian Deng 1 Introduction We started to talk about variational

More information

STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 3 Linear

More information

14 : Theory of Variational Inference: Inner and Outer Approximation

14 : Theory of Variational Inference: Inner and Outer Approximation 10-708: Probabilistic Graphical Models 10-708, Spring 2014 14 : Theory of Variational Inference: Inner and Outer Approximation Lecturer: Eric P. Xing Scribes: Yu-Hsin Kuo, Amos Ng 1 Introduction Last lecture

More information

Lecture 21: Spectral Learning for Graphical Models

Lecture 21: Spectral Learning for Graphical Models 10-708: Probabilistic Graphical Models 10-708, Spring 2016 Lecture 21: Spectral Learning for Graphical Models Lecturer: Eric P. Xing Scribes: Maruan Al-Shedivat, Wei-Cheng Chang, Frederick Liu 1 Motivation

More information

Probabilistic Graphical Models (I)

Probabilistic Graphical Models (I) Probabilistic Graphical Models (I) Hongxin Zhang zhx@cad.zju.edu.cn State Key Lab of CAD&CG, ZJU 2015-03-31 Probabilistic Graphical Models Modeling many real-world problems => a large number of random

More information

Lecture 9: PGM Learning

Lecture 9: PGM Learning 13 Oct 2014 Intro. to Stats. Machine Learning COMP SCI 4401/7401 Table of Contents I Learning parameters in MRFs 1 Learning parameters in MRFs Inference and Learning Given parameters (of potentials) and

More information

13 : Variational Inference: Loopy Belief Propagation and Mean Field

13 : Variational Inference: Loopy Belief Propagation and Mean Field 10-708: Probabilistic Graphical Models 10-708, Spring 2012 13 : Variational Inference: Loopy Belief Propagation and Mean Field Lecturer: Eric P. Xing Scribes: Peter Schulam and William Wang 1 Introduction

More information

CS242: Probabilistic Graphical Models Lecture 4B: Learning Tree-Structured and Directed Graphs

CS242: Probabilistic Graphical Models Lecture 4B: Learning Tree-Structured and Directed Graphs CS242: Probabilistic Graphical Models Lecture 4B: Learning Tree-Structured and Directed Graphs Professor Erik Sudderth Brown University Computer Science October 6, 2016 Some figures and materials courtesy

More information

Introduction to Machine Learning

Introduction to Machine Learning Introduction to Machine Learning Brown University CSCI 1950-F, Spring 2012 Prof. Erik Sudderth Lecture 20: Expectation Maximization Algorithm EM for Mixture Models Many figures courtesy Kevin Murphy s

More information

A graph contains a set of nodes (vertices) connected by links (edges or arcs)

A graph contains a set of nodes (vertices) connected by links (edges or arcs) BOLTZMANN MACHINES Generative Models Graphical Models A graph contains a set of nodes (vertices) connected by links (edges or arcs) In a probabilistic graphical model, each node represents a random variable,

More information

Junction Tree, BP and Variational Methods

Junction Tree, BP and Variational Methods Junction Tree, BP and Variational Methods Adrian Weller MLSALT4 Lecture Feb 21, 2018 With thanks to David Sontag (MIT) and Tony Jebara (Columbia) for use of many slides and illustrations For more information,

More information

Bayesian Machine Learning - Lecture 7

Bayesian Machine Learning - Lecture 7 Bayesian Machine Learning - Lecture 7 Guido Sanguinetti Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh gsanguin@inf.ed.ac.uk March 4, 2015 Today s lecture 1

More information

Probabilistic Graphical Models

Probabilistic Graphical Models 2016 Robert Nowak Probabilistic Graphical Models 1 Introduction We have focused mainly on linear models for signals, in particular the subspace model x = Uθ, where U is a n k matrix and θ R k is a vector

More information

Lecture 16 Deep Neural Generative Models

Lecture 16 Deep Neural Generative Models Lecture 16 Deep Neural Generative Models CMSC 35246: Deep Learning Shubhendu Trivedi & Risi Kondor University of Chicago May 22, 2017 Approach so far: We have considered simple models and then constructed

More information

Introduction to Graphical Models

Introduction to Graphical Models Introduction to Graphical Models The 15 th Winter School of Statistical Physics POSCO International Center & POSTECH, Pohang 2018. 1. 9 (Tue.) Yung-Kyun Noh GENERALIZATION FOR PREDICTION 2 Probabilistic

More information

Statistical Approaches to Learning and Discovery

Statistical Approaches to Learning and Discovery Statistical Approaches to Learning and Discovery Graphical Models Zoubin Ghahramani & Teddy Seidenfeld zoubin@cs.cmu.edu & teddy@stat.cmu.edu CALD / CS / Statistics / Philosophy Carnegie Mellon University

More information

Undirected Graphical Models: Markov Random Fields

Undirected Graphical Models: Markov Random Fields Undirected Graphical Models: Markov Random Fields 40-956 Advanced Topics in AI: Probabilistic Graphical Models Sharif University of Technology Soleymani Spring 2015 Markov Random Field Structure: undirected

More information

Deep Learning Srihari. Deep Belief Nets. Sargur N. Srihari

Deep Learning Srihari. Deep Belief Nets. Sargur N. Srihari Deep Belief Nets Sargur N. Srihari srihari@cedar.buffalo.edu Topics 1. Boltzmann machines 2. Restricted Boltzmann machines 3. Deep Belief Networks 4. Deep Boltzmann machines 5. Boltzmann machines for continuous

More information

STA 414/2104: Machine Learning

STA 414/2104: Machine Learning STA 414/2104: Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistics! rsalakhu@cs.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 9 Sequential Data So far

More information

Graphical Models for Collaborative Filtering

Graphical Models for Collaborative Filtering Graphical Models for Collaborative Filtering Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Sequence modeling HMM, Kalman Filter, etc.: Similarity: the same graphical model topology,

More information

Variational Inference (11/04/13)

Variational Inference (11/04/13) STA561: Probabilistic machine learning Variational Inference (11/04/13) Lecturer: Barbara Engelhardt Scribes: Matt Dickenson, Alireza Samany, Tracy Schifeling 1 Introduction In this lecture we will further

More information

Machine Learning for Data Science (CS4786) Lecture 24

Machine Learning for Data Science (CS4786) Lecture 24 Machine Learning for Data Science (CS4786) Lecture 24 Graphical Models: Approximate Inference Course Webpage : http://www.cs.cornell.edu/courses/cs4786/2016sp/ BELIEF PROPAGATION OR MESSAGE PASSING Each

More information

Chris Bishop s PRML Ch. 8: Graphical Models

Chris Bishop s PRML Ch. 8: Graphical Models Chris Bishop s PRML Ch. 8: Graphical Models January 24, 2008 Introduction Visualize the structure of a probabilistic model Design and motivate new models Insights into the model s properties, in particular

More information

Probabilistic Graphical Models. Theory of Variational Inference: Inner and Outer Approximation. Lecture 15, March 4, 2013

Probabilistic Graphical Models. Theory of Variational Inference: Inner and Outer Approximation. Lecture 15, March 4, 2013 School of Computer Science Probabilistic Graphical Models Theory of Variational Inference: Inner and Outer Approximation Junming Yin Lecture 15, March 4, 2013 Reading: W & J Book Chapters 1 Roadmap Two

More information

Pattern Recognition and Machine Learning

Pattern Recognition and Machine Learning Christopher M. Bishop Pattern Recognition and Machine Learning ÖSpri inger Contents Preface Mathematical notation Contents vii xi xiii 1 Introduction 1 1.1 Example: Polynomial Curve Fitting 4 1.2 Probability

More information

Sequence labeling. Taking collective a set of interrelated instances x 1,, x T and jointly labeling them

Sequence labeling. Taking collective a set of interrelated instances x 1,, x T and jointly labeling them HMM, MEMM and CRF 40-957 Special opics in Artificial Intelligence: Probabilistic Graphical Models Sharif University of echnology Soleymani Spring 2014 Sequence labeling aking collective a set of interrelated

More information

Linear Dynamical Systems

Linear Dynamical Systems Linear Dynamical Systems Sargur N. srihari@cedar.buffalo.edu Machine Learning Course: http://www.cedar.buffalo.edu/~srihari/cse574/index.html Two Models Described by Same Graph Latent variables Observations

More information

Inference in Graphical Models Variable Elimination and Message Passing Algorithm

Inference in Graphical Models Variable Elimination and Message Passing Algorithm Inference in Graphical Models Variable Elimination and Message Passing lgorithm Le Song Machine Learning II: dvanced Topics SE 8803ML, Spring 2012 onditional Independence ssumptions Local Markov ssumption

More information

Bayesian Networks: Construction, Inference, Learning and Causal Interpretation. Volker Tresp Summer 2014

Bayesian Networks: Construction, Inference, Learning and Causal Interpretation. Volker Tresp Summer 2014 Bayesian Networks: Construction, Inference, Learning and Causal Interpretation Volker Tresp Summer 2014 1 Introduction So far we were mostly concerned with supervised learning: we predicted one or several

More information

Inference in Bayesian Networks

Inference in Bayesian Networks Andrea Passerini passerini@disi.unitn.it Machine Learning Inference in graphical models Description Assume we have evidence e on the state of a subset of variables E in the model (i.e. Bayesian Network)

More information

9 Forward-backward algorithm, sum-product on factor graphs

9 Forward-backward algorithm, sum-product on factor graphs Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms For Inference Fall 2014 9 Forward-backward algorithm, sum-product on factor graphs The previous

More information

Lecture 15. Probabilistic Models on Graph

Lecture 15. Probabilistic Models on Graph Lecture 15. Probabilistic Models on Graph Prof. Alan Yuille Spring 2014 1 Introduction We discuss how to define probabilistic models that use richly structured probability distributions and describe how

More information

Bayesian Networks: Construction, Inference, Learning and Causal Interpretation. Volker Tresp Summer 2016

Bayesian Networks: Construction, Inference, Learning and Causal Interpretation. Volker Tresp Summer 2016 Bayesian Networks: Construction, Inference, Learning and Causal Interpretation Volker Tresp Summer 2016 1 Introduction So far we were mostly concerned with supervised learning: we predicted one or several

More information

Recent Advances in Bayesian Inference Techniques

Recent Advances in Bayesian Inference Techniques Recent Advances in Bayesian Inference Techniques Christopher M. Bishop Microsoft Research, Cambridge, U.K. research.microsoft.com/~cmbishop SIAM Conference on Data Mining, April 2004 Abstract Bayesian

More information

ECE521 Tutorial 11. Topic Review. ECE521 Winter Credits to Alireza Makhzani, Alex Schwing, Rich Zemel and TAs for slides. ECE521 Tutorial 11 / 4

ECE521 Tutorial 11. Topic Review. ECE521 Winter Credits to Alireza Makhzani, Alex Schwing, Rich Zemel and TAs for slides. ECE521 Tutorial 11 / 4 ECE52 Tutorial Topic Review ECE52 Winter 206 Credits to Alireza Makhzani, Alex Schwing, Rich Zemel and TAs for slides ECE52 Tutorial ECE52 Winter 206 Credits to Alireza / 4 Outline K-means, PCA 2 Bayesian

More information

Graphical Models for Statistical Inference and Data Assimilation

Graphical Models for Statistical Inference and Data Assimilation Graphical Models for Statistical Inference and Data Assimilation Alexander T. Ihler a Sergey Kirshner a Michael Ghil b,c Andrew W. Robertson d Padhraic Smyth a a Donald Bren School of Information and Computer

More information

Walk-Sum Interpretation and Analysis of Gaussian Belief Propagation

Walk-Sum Interpretation and Analysis of Gaussian Belief Propagation Walk-Sum Interpretation and Analysis of Gaussian Belief Propagation Jason K. Johnson, Dmitry M. Malioutov and Alan S. Willsky Department of Electrical Engineering and Computer Science Massachusetts Institute

More information

Bayesian Learning in Undirected Graphical Models

Bayesian Learning in Undirected Graphical Models Bayesian Learning in Undirected Graphical Models Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London, UK http://www.gatsby.ucl.ac.uk/ Work with: Iain Murray and Hyun-Chul

More information

Graphical Models for Statistical Inference and Data Assimilation

Graphical Models for Statistical Inference and Data Assimilation Graphical Models for Statistical Inference and Data Assimilation Alexander T. Ihler a Sergey Kirshner b Michael Ghil c,d Andrew W. Robertson e Padhraic Smyth a a Donald Bren School of Information and Computer

More information

Machine Learning Techniques for Computer Vision

Machine Learning Techniques for Computer Vision Machine Learning Techniques for Computer Vision Part 2: Unsupervised Learning Microsoft Research Cambridge x 3 1 0.5 0.2 0 0.5 0.3 0 0.5 1 ECCV 2004, Prague x 2 x 1 Overview of Part 2 Mixture models EM

More information

Probabilistic Graphical Models

Probabilistic Graphical Models School of Computer Science Probabilistic Graphical Models Variational Inference IV: Variational Principle II Junming Yin Lecture 17, March 21, 2012 X 1 X 1 X 1 X 1 X 2 X 3 X 2 X 2 X 3 X 3 Reading: X 4

More information

Probabilistic Graphical Models

Probabilistic Graphical Models School of Computer Science Probabilistic Graphical Models Variational Inference II: Mean Field Method and Variational Principle Junming Yin Lecture 15, March 7, 2012 X 1 X 1 X 1 X 1 X 2 X 3 X 2 X 2 X 3

More information

Graphical Models and Kernel Methods

Graphical Models and Kernel Methods Graphical Models and Kernel Methods Jerry Zhu Department of Computer Sciences University of Wisconsin Madison, USA MLSS June 17, 2014 1 / 123 Outline Graphical Models Probabilistic Inference Directed vs.

More information

13 : Variational Inference: Loopy Belief Propagation

13 : Variational Inference: Loopy Belief Propagation 10-708: Probabilistic Graphical Models 10-708, Spring 2014 13 : Variational Inference: Loopy Belief Propagation Lecturer: Eric P. Xing Scribes: Rajarshi Das, Zhengzhong Liu, Dishan Gupta 1 Introduction

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Lecture Notes Fall 2009 November, 2009 Byoung-Ta Zhang School of Computer Science and Engineering & Cognitive Science, Brain Science, and Bioinformatics Seoul National University

More information

Graphical models for statistical inference and data assimilation

Graphical models for statistical inference and data assimilation Physica D ( ) www.elsevier.com/locate/physd Graphical models for statistical inference and data assimilation Alexander T. Ihler a,, Sergey Kirshner b, Michael Ghil c,d, Andrew W. Robertson e, Padhraic

More information

Undirected Graphical Models

Undirected Graphical Models Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Properties Properties 3 Generative vs. Conditional

More information

Lecture 13 : Variational Inference: Mean Field Approximation

Lecture 13 : Variational Inference: Mean Field Approximation 10-708: Probabilistic Graphical Models 10-708, Spring 2017 Lecture 13 : Variational Inference: Mean Field Approximation Lecturer: Willie Neiswanger Scribes: Xupeng Tong, Minxing Liu 1 Problem Setup 1.1

More information

Message Passing Algorithms and Junction Tree Algorithms

Message Passing Algorithms and Junction Tree Algorithms Message Passing lgorithms and Junction Tree lgorithms Le Song Machine Learning II: dvanced Topics S 8803ML, Spring 2012 Inference in raphical Models eneral form of the inference problem P X 1,, X n Ψ(

More information

CS242: Probabilistic Graphical Models Lecture 7B: Markov Chain Monte Carlo & Gibbs Sampling

CS242: Probabilistic Graphical Models Lecture 7B: Markov Chain Monte Carlo & Gibbs Sampling CS242: Probabilistic Graphical Models Lecture 7B: Markov Chain Monte Carlo & Gibbs Sampling Professor Erik Sudderth Brown University Computer Science October 27, 2016 Some figures and materials courtesy

More information

Generative and Discriminative Approaches to Graphical Models CMSC Topics in AI

Generative and Discriminative Approaches to Graphical Models CMSC Topics in AI Generative and Discriminative Approaches to Graphical Models CMSC 35900 Topics in AI Lecture 2 Yasemin Altun January 26, 2007 Review of Inference on Graphical Models Elimination algorithm finds single

More information

Learning MN Parameters with Alternative Objective Functions. Sargur Srihari

Learning MN Parameters with Alternative Objective Functions. Sargur Srihari Learning MN Parameters with Alternative Objective Functions Sargur srihari@cedar.buffalo.edu 1 Topics Max Likelihood & Contrastive Objectives Contrastive Objective Learning Methods Pseudo-likelihood Gradient

More information

Probabilistic Graphical Models: Representation and Inference

Probabilistic Graphical Models: Representation and Inference Probabilistic Graphical Models: Representation and Inference Aaron C. Courville Université de Montréal Note: Material for the slides is taken directly from a presentation prepared by Andrew Moore 1 Overview

More information

STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 7 Approximate

More information

12 : Variational Inference I

12 : Variational Inference I 10-708: Probabilistic Graphical Models, Spring 2015 12 : Variational Inference I Lecturer: Eric P. Xing Scribes: Fattaneh Jabbari, Eric Lei, Evan Shapiro 1 Introduction Probabilistic inference is one of

More information

Variational Inference. Sargur Srihari

Variational Inference. Sargur Srihari Variational Inference Sargur srihari@cedar.buffalo.edu 1 Plan of discussion We first describe inference with PGMs and the intractability of exact inference Then give a taxonomy of inference algorithms

More information

Variational Autoencoders

Variational Autoencoders Variational Autoencoders Recap: Story so far A classification MLP actually comprises two components A feature extraction network that converts the inputs into linearly separable features Or nearly linearly

More information

Bayesian Learning in Undirected Graphical Models

Bayesian Learning in Undirected Graphical Models Bayesian Learning in Undirected Graphical Models Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London, UK http://www.gatsby.ucl.ac.uk/ and Center for Automated Learning and

More information

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Algorithms For Inference Fall 2014

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Algorithms For Inference Fall 2014 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.438 Algorithms For Inference Fall 2014 Problem Set 3 Issued: Thursday, September 25, 2014 Due: Thursday,

More information

An Introduction to Bayesian Machine Learning

An Introduction to Bayesian Machine Learning 1 An Introduction to Bayesian Machine Learning José Miguel Hernández-Lobato Department of Engineering, Cambridge University April 8, 2013 2 What is Machine Learning? The design of computational systems

More information

Factor Analysis and Kalman Filtering (11/2/04)

Factor Analysis and Kalman Filtering (11/2/04) CS281A/Stat241A: Statistical Learning Theory Factor Analysis and Kalman Filtering (11/2/04) Lecturer: Michael I. Jordan Scribes: Byung-Gon Chun and Sunghoon Kim 1 Factor Analysis Factor analysis is used

More information

CS242: Probabilistic Graphical Models Lecture 4A: MAP Estimation & Graph Structure Learning

CS242: Probabilistic Graphical Models Lecture 4A: MAP Estimation & Graph Structure Learning CS242: Probabilistic Graphical Models Lecture 4A: MAP Estimation & Graph Structure Learning Professor Erik Sudderth Brown University Computer Science October 4, 2016 Some figures and materials courtesy

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Lecture 10 Undirected Models CS/CNS/EE 155 Andreas Krause Announcements Homework 2 due this Wednesday (Nov 4) in class Project milestones due next Monday (Nov 9) About half

More information

Sampling Algorithms for Probabilistic Graphical models

Sampling Algorithms for Probabilistic Graphical models Sampling Algorithms for Probabilistic Graphical models Vibhav Gogate University of Washington References: Chapter 12 of Probabilistic Graphical models: Principles and Techniques by Daphne Koller and Nir

More information

6.047 / Computational Biology: Genomes, Networks, Evolution Fall 2008

6.047 / Computational Biology: Genomes, Networks, Evolution Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 6.047 / 6.878 Computational Biology: Genomes, Networks, Evolution Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

More information

Parameter learning in CRF s

Parameter learning in CRF s Parameter learning in CRF s June 01, 2009 Structured output learning We ish to learn a discriminant (or compatability) function: F : X Y R (1) here X is the space of inputs and Y is the space of outputs.

More information

Based on slides by Richard Zemel

Based on slides by Richard Zemel CSC 412/2506 Winter 2018 Probabilistic Learning and Reasoning Lecture 3: Directed Graphical Models and Latent Variables Based on slides by Richard Zemel Learning outcomes What aspects of a model can we

More information

Machine Learning Lecture 14

Machine Learning Lecture 14 Many slides adapted from B. Schiele, S. Roth, Z. Gharahmani Machine Learning Lecture 14 Undirected Graphical Models & Inference 23.06.2015 Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de/ leibe@vision.rwth-aachen.de

More information

Lecture 7: Con3nuous Latent Variable Models

Lecture 7: Con3nuous Latent Variable Models CSC2515 Fall 2015 Introduc3on to Machine Learning Lecture 7: Con3nuous Latent Variable Models All lecture slides will be available as.pdf on the course website: http://www.cs.toronto.edu/~urtasun/courses/csc2515/

More information

Probabilistic and Bayesian Machine Learning

Probabilistic and Bayesian Machine Learning Probabilistic and Bayesian Machine Learning Day 4: Expectation and Belief Propagation Yee Whye Teh ywteh@gatsby.ucl.ac.uk Gatsby Computational Neuroscience Unit University College London http://www.gatsby.ucl.ac.uk/

More information

Approximate Inference Part 1 of 2

Approximate Inference Part 1 of 2 Approximate Inference Part 1 of 2 Tom Minka Microsoft Research, Cambridge, UK Machine Learning Summer School 2009 http://mlg.eng.cam.ac.uk/mlss09/ Bayesian paradigm Consistent use of probability theory

More information

Expectation Propagation Algorithm

Expectation Propagation Algorithm Expectation Propagation Algorithm 1 Shuang Wang School of Electrical and Computer Engineering University of Oklahoma, Tulsa, OK, 74135 Email: {shuangwang}@ou.edu This note contains three parts. First,

More information

Approximate Inference Part 1 of 2

Approximate Inference Part 1 of 2 Approximate Inference Part 1 of 2 Tom Minka Microsoft Research, Cambridge, UK Machine Learning Summer School 2009 http://mlg.eng.cam.ac.uk/mlss09/ 1 Bayesian paradigm Consistent use of probability theory

More information

3 : Representation of Undirected GM

3 : Representation of Undirected GM 10-708: Probabilistic Graphical Models 10-708, Spring 2016 3 : Representation of Undirected GM Lecturer: Eric P. Xing Scribes: Longqi Cai, Man-Chia Chang 1 MRF vs BN There are two types of graphical models:

More information

Conditional Random Fields and beyond DANIEL KHASHABI CS 546 UIUC, 2013

Conditional Random Fields and beyond DANIEL KHASHABI CS 546 UIUC, 2013 Conditional Random Fields and beyond DANIEL KHASHABI CS 546 UIUC, 2013 Outline Modeling Inference Training Applications Outline Modeling Problem definition Discriminative vs. Generative Chain CRF General

More information

Directed and Undirected Graphical Models

Directed and Undirected Graphical Models Directed and Undirected Davide Bacciu Dipartimento di Informatica Università di Pisa bacciu@di.unipi.it Machine Learning: Neural Networks and Advanced Models (AA2) Last Lecture Refresher Lecture Plan Directed

More information

Embedded Trees: Estimation of Gaussian Processes on Graphs with Cycles

Embedded Trees: Estimation of Gaussian Processes on Graphs with Cycles SUBMIED O IEEE RANSACIONS ON SIGNAL PROCESSING 1 Embedded rees: Estimation of Gaussian Processes on Graphs with Cycles Erik B. Sudderth *, Martin J. Wainwright, and Alan S. Willsky EDICS: 2-HIER (HIERARCHICAL

More information

CS281A/Stat241A Lecture 19

CS281A/Stat241A Lecture 19 CS281A/Stat241A Lecture 19 p. 1/4 CS281A/Stat241A Lecture 19 Junction Tree Algorithm Peter Bartlett CS281A/Stat241A Lecture 19 p. 2/4 Announcements My office hours: Tuesday Nov 3 (today), 1-2pm, in 723

More information

CS 2750: Machine Learning. Bayesian Networks. Prof. Adriana Kovashka University of Pittsburgh March 14, 2016

CS 2750: Machine Learning. Bayesian Networks. Prof. Adriana Kovashka University of Pittsburgh March 14, 2016 CS 2750: Machine Learning Bayesian Networks Prof. Adriana Kovashka University of Pittsburgh March 14, 2016 Plan for today and next week Today and next time: Bayesian networks (Bishop Sec. 8.1) Conditional

More information

Lecture 6: Graphical Models

Lecture 6: Graphical Models Lecture 6: Graphical Models Kai-Wei Chang CS @ Uniersity of Virginia kw@kwchang.net Some slides are adapted from Viek Skirmar s course on Structured Prediction 1 So far We discussed sequence labeling tasks:

More information

Learning Parameters of Undirected Models. Sargur Srihari

Learning Parameters of Undirected Models. Sargur Srihari Learning Parameters of Undirected Models Sargur srihari@cedar.buffalo.edu 1 Topics Difficulties due to Global Normalization Likelihood Function Maximum Likelihood Parameter Estimation Simple and Conjugate

More information

Exact Inference I. Mark Peot. In this lecture we will look at issues associated with exact inference. = =

Exact Inference I. Mark Peot. In this lecture we will look at issues associated with exact inference. = = Exact Inference I Mark Peot In this lecture we will look at issues associated with exact inference 10 Queries The objective of probabilistic inference is to compute a joint distribution of a set of query

More information

Clustering with k-means and Gaussian mixture distributions

Clustering with k-means and Gaussian mixture distributions Clustering with k-means and Gaussian mixture distributions Machine Learning and Object Recognition 2017-2018 Jakob Verbeek Clustering Finding a group structure in the data Data in one cluster similar to

More information

Learning MN Parameters with Approximation. Sargur Srihari

Learning MN Parameters with Approximation. Sargur Srihari Learning MN Parameters with Approximation Sargur srihari@cedar.buffalo.edu 1 Topics Iterative exact learning of MN parameters Difficulty with exact methods Approximate methods Approximate Inference Belief

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models Lecture 9: Variational Inference Relaxations Volkan Cevher, Matthias Seeger Ecole Polytechnique Fédérale de Lausanne 24/10/2011 (EPFL) Graphical Models 24/10/2011 1 / 15

More information

Probabilistic Graphical Models: MRFs and CRFs. CSE628: Natural Language Processing Guest Lecturer: Veselin Stoyanov

Probabilistic Graphical Models: MRFs and CRFs. CSE628: Natural Language Processing Guest Lecturer: Veselin Stoyanov Probabilistic Graphical Models: MRFs and CRFs CSE628: Natural Language Processing Guest Lecturer: Veselin Stoyanov Why PGMs? PGMs can model joint probabilities of many events. many techniques commonly

More information

Learning Parameters of Undirected Models. Sargur Srihari

Learning Parameters of Undirected Models. Sargur Srihari Learning Parameters of Undirected Models Sargur srihari@cedar.buffalo.edu 1 Topics Log-linear Parameterization Likelihood Function Maximum Likelihood Parameter Estimation Simple and Conjugate Gradient

More information

Lecture 4: Probabilistic Learning. Estimation Theory. Classification with Probability Distributions

Lecture 4: Probabilistic Learning. Estimation Theory. Classification with Probability Distributions DD2431 Autumn, 2014 1 2 3 Classification with Probability Distributions Estimation Theory Classification in the last lecture we assumed we new: P(y) Prior P(x y) Lielihood x2 x features y {ω 1,..., ω K

More information

Representation of undirected GM. Kayhan Batmanghelich

Representation of undirected GM. Kayhan Batmanghelich Representation of undirected GM Kayhan Batmanghelich Review Review: Directed Graphical Model Represent distribution of the form ny p(x 1,,X n = p(x i (X i i=1 Factorizes in terms of local conditional probabilities

More information

Embedded Trees: Estimation of Gaussian Processes on Graphs with Cycles

Embedded Trees: Estimation of Gaussian Processes on Graphs with Cycles MI LIDS ECHNICAL REPOR P-2562 1 Embedded rees: Estimation of Gaussian Processes on Graphs with Cycles Erik B. Sudderth, Martin J. Wainwright, and Alan S. Willsky MI LIDS ECHNICAL REPOR P-2562 (APRIL 2003)

More information

Markov Networks.

Markov Networks. Markov Networks www.biostat.wisc.edu/~dpage/cs760/ Goals for the lecture you should understand the following concepts Markov network syntax Markov network semantics Potential functions Partition function

More information