Graphical models and causality: Directed acyclic graphs (DAGs) and conditional (in)dependence
|
|
- Erika Bell
- 5 years ago
- Views:
Transcription
1 Graphical models and causality: Directed acyclic graphs (DAGs) and conditional (in)dependence
2 General overview Introduction Directed acyclic graphs (DAGs) and conditional independence DAGs and causal effects Learning DAGs from observational data IDA algorithm Further problems 2
3 Overview DAGs and conditional (in)dependence Directed acyclic graphs Factorization of the joint density Markov property d-separation 3
4 Graph terminology A graph G = V, E consists of vertices (nodes) V and edges E There is at most one edge between every pair of vertices Two vertices are adjacent if there is an edge between them If all edges are directed (i j), the graph is called directed A path between i and j is a sequence of distinct vertices (i,, j) such that successive vertices are adjacent A directed path from i to j is a path between i and j where all edges are pointing towards j, i.e., i j. A non-directed path from i to j is a path from i to j that is not directed. A cycle is a path (i, j,, k) plus an edge between k and i A directed cycle is a directed path (i, j,, k) from i to k, plus an edge k i A directed acyclic graph (DAG) is a directed graph without directed cycles 4
5 Example Which of the following are (directed) paths? (4,2,1,3) (3,4,2,1,4,5) (3,4,2,1) 3,4,5 a directed path from 3 to 5 (1,2,4,1) is a cycle, but not a directed cycle This graph is a DAG not a path: 1 and 3 are not adjacent not a path: vertices are not distinct a path, but not a directed path Adding the edge 5 2 yields a directed cycle 2,4,5,2. The graph is no longer a DAG. 5
6 Graph terminology If i j, then i is a parent of j, and j is a child of i If there is a directed path from i to j, then i is an ancestor of j and j is a descendant of i. Each vertex is also an ancestor and descendant of itself. The sets of parents, children, descendants and ancestors of i in G are denoted by pa(i, G), ch(i, G), desc(i, G), an i, G. We omit G if the graph is clear from the context. We write sets of variables in bold face All definitions are applied disjunctively to sets. Example: pa S = k S pa(k) The non-descendants of S are the complement of desc S : nondesc S V desc(s) 6
7 Example is a parent of 4, and 4 is a child of 1 1 is an ancestor of 5, and 5 is a descendant of 1 ch(4) = {5} desc(4) = {4,5} pa(4) = {1,2,3} an(4) = {1,2,3,4} pa 2,4 = 1 1,2,3 = 1,2,3 desc 2,3 = 2,4,5 3,4,5 = 2,3,4,5 nondesc 2,3 = 1,2,3,4,5 2,3,4,5 = {1} 7
8 DAGs and random variables Each vertex represents a random variable: we use i to denote both the vertex i and the random variable X i An edge denotes a relationship between the pair of variables (we will make this more precise later) 8
9 Factorization of the joint density Suppose we have p binary random variables. Then specifying the joint distribution requires a probability table of size 2 p. Can we do this more efficiently? We always have: f x 1,, x p = f x 1 f x 2 x 1 f(x p x 1, x p 1 ) A set of variables pa(j) is said to be the Markovian parents of X j if it is a minimal subset of {X 1, X j 1 } such that f x j x 1,, x j 1 = f(x j pa j ). p Then f x 1,, x p = j=1 f(x j pa j ) We can draw a DAG accordingly 9
10 Examples (X 1, X 2, X 3 ) and X 1 X 3 X 2 is only (conditional) independence: f x 1 x 2, x 3 = f x 1 x 2 and f x 3 x 1, x 2 = f(x 3 x 2 ) Then Or Or f x 1, x 2, x 3 = f x 1 f x 2 x 1 f x 3 x 1, x 2 = f x 1 f x 2 x 1 f x 3 x 2 DAG: f x 3, x 2, x 1 = f x 3 f x 2 x 3 f x 1 x 2, x 3 = f x 3 f x 2 x 3 f x 1 x 2 DAG: f x 1, x 3, x 2 = f x 1 f x 3 x 1 f x 2 x 1, x 3 DAG:
11 Example First order Markov chain: f X 1,, X p = f X 1 f X 2 X 1 f X p X 1,, X p 1 = f X 1 f X 2 X 1 f(x p X p 1 ) DAG: 1 2 p 11
12 Bayesian network A Bayesian network is pair (G, f), where f factorizes according to G: p f x 1,, x p = f(x j pa j, G ) j=1 Bayesian networks can be used for: Estimation of the joint density from lower order conditional densities (if the parent sets are small) Reading off conditional independencies from the DAG Probabilistic reasoning (expert systems) Causal inference 12
13 Bayesian network A Bayesian network is pair (G, f), where f factorizes according to G: p f x 1,, x p = f(x j pa j, G ) j=1 Bayesian networks can be used for: Estimation of the joint density from low order conditional densities (if the parent sets are small) Reading off conditional independencies from the DAG Probabilistic reasoning (expert systems) Causal inference 13
14 Reading off conditional independencies: Markov property Markov model: 1 2 t 1 t t + 1. The future is independent of the past given the present: X t+1 X t 1, X t 2,, X 1 X t In Bayesian networks, we have a similar Markov property. Let S be any collection of nodes. Then S is independent of its nondescendants given its parents: S nondesc S pa S pa(s) 14
15 Example Markov property smoking yellow teeth tar in lungs asbestos cancer Note: pa yellow teeth = smoking cancer nondesc(yellow teeth) Hence, yellow teeth cancer smoking in any distribution that factorizes according to this DAG 15
16 Graph terminology The skeleton of a graph is the graph obtained by removing all arrowheads A node i is a collider on a path if the path contains i (the arrows collide at i). Otherwise, it is a non-collider on the path Examples: 4 is a collider on the path (3,4,1) 4 is a non-collider on the path (3,4,5) Collider status depends is relative to a path 16
17 Reading off conditional independencies: d-separation The Markov property cannot be used to read off arbitrary conditional (in)dependencies. For this we have d-separation. A path between i to j is blocked by a set S if at least one of the following holds: There is a non-collider on the path that is in S; or There is a collider on the path such that neither this collider nor any of its descendants are in S. A path that is not blocked is d-connecting. If all paths between i to j are blocked by S, then i and j are d- separated by S. Otherwise they are d-connected given S. In any distribution that factorizes according to a DAG: if i and j are d-separated by S in the DAG, then X i and X j are conditionally independent given S in the distribution 17
18 Examples d-separation smoking yellow teeth tar in lungs asbestos cancer Denote d-separation by. Which of the following hold? yellow teeth cancer smoking tar asbestos tar asbestos cancer yellow teeth asbestos cancer yes yes no no 18
19 Bayesian network A Bayesian network is pair (G, f), where f factorizes according to G: p f x 1,, x p = f(x j pa j, G ) j=1 Bayesian networks can be used for: Estimation of the joint density from low order conditional densities (if the parent sets are small) Reading off conditional independencies from the DAG Probabilistic reasoning (expert systems); see R-code Causal inference 19
20 Bayesian network A Bayesian network is pair (G, f), where f factorizes according to G: p f x 1,, x p = f(x j pa j, G ) j=1 Bayesian networks can be used for: Estimation of the joint density from low order conditional densities (if the parent sets are small) Reading off conditional independencies from the DAG Probabilistic reasoning (expert systems) Causal inference: next topic 20
1. what conditional independencies are implied by the graph. 2. whether these independecies correspond to the probability distribution
NETWORK ANALYSIS Lourens Waldorp PROBABILITY AND GRAPHS The objective is to obtain a correspondence between the intuitive pictures (graphs) of variables of interest and the probability distributions of
More informationIntroduction to Causal Calculus
Introduction to Causal Calculus Sanna Tyrväinen University of British Columbia August 1, 2017 1 / 1 2 / 1 Bayesian network Bayesian networks are Directed Acyclic Graphs (DAGs) whose nodes represent random
More informationProbabilistic 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 informationCausal Inference & Reasoning with Causal Bayesian Networks
Causal Inference & Reasoning with Causal Bayesian Networks Neyman-Rubin Framework Potential Outcome Framework: for each unit k and each treatment i, there is a potential outcome on an attribute U, U ik,
More informationRapid Introduction to Machine Learning/ Deep Learning
Rapid Introduction to Machine Learning/ Deep Learning Hyeong In Choi Seoul National University 1/32 Lecture 5a Bayesian network April 14, 2016 2/32 Table of contents 1 1. Objectives of Lecture 5a 2 2.Bayesian
More informationDirected Graphical Models
CS 2750: Machine Learning Directed Graphical Models Prof. Adriana Kovashka University of Pittsburgh March 28, 2017 Graphical Models If no assumption of independence is made, must estimate an exponential
More informationRecall from last time. Lecture 3: Conditional independence and graph structure. Example: A Bayesian (belief) network.
ecall from last time Lecture 3: onditional independence and graph structure onditional independencies implied by a belief network Independence maps (I-maps) Factorization theorem The Bayes ball algorithm
More informationCausality in Econometrics (3)
Graphical Causal Models References Causality in Econometrics (3) Alessio Moneta Max Planck Institute of Economics Jena moneta@econ.mpg.de 26 April 2011 GSBC Lecture Friedrich-Schiller-Universität Jena
More informationCS 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 informationMarkov properties for directed graphs
Graphical Models, Lecture 7, Michaelmas Term 2009 November 2, 2009 Definitions Structural relations among Markov properties Factorization G = (V, E) simple undirected graph; σ Say σ satisfies (P) the pairwise
More informationProbabilistic Graphical Networks: Definitions and Basic Results
This document gives a cursory overview of Probabilistic Graphical Networks. The material has been gleaned from different sources. I make no claim to original authorship of this material. Bayesian Graphical
More informationPart I. C. M. Bishop PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS
Part I C. M. Bishop PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS Probabilistic Graphical Models Graphical representation of a probabilistic model Each variable corresponds to a
More informationMotivation. Bayesian Networks in Epistemology and Philosophy of Science Lecture. Overview. Organizational Issues
Bayesian Networks in Epistemology and Philosophy of Science Lecture 1: Bayesian Networks Center for Logic and Philosophy of Science Tilburg University, The Netherlands Formal Epistemology Course Northern
More informationCS Lecture 3. More Bayesian Networks
CS 6347 Lecture 3 More Bayesian Networks Recap Last time: Complexity challenges Representing distributions Computing probabilities/doing inference Introduction to Bayesian networks Today: D-separation,
More informationLearning in Bayesian Networks
Learning in Bayesian Networks Florian Markowetz Max-Planck-Institute for Molecular Genetics Computational Molecular Biology Berlin Berlin: 20.06.2002 1 Overview 1. Bayesian Networks Stochastic Networks
More informationChris 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 informationIdentifiability assumptions for directed graphical models with feedback
Biometrika, pp. 1 26 C 212 Biometrika Trust Printed in Great Britain Identifiability assumptions for directed graphical models with feedback BY GUNWOONG PARK Department of Statistics, University of Wisconsin-Madison,
More informationPreliminaries Bayesian Networks Graphoid Axioms d-separation Wrap-up. Bayesian Networks. Brandon Malone
Preliminaries Graphoid Axioms d-separation Wrap-up Much of this material is adapted from Chapter 4 of Darwiche s book January 23, 2014 Preliminaries Graphoid Axioms d-separation Wrap-up 1 Preliminaries
More informationDirected 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 informationData Mining 2018 Bayesian Networks (1)
Data Mining 2018 Bayesian Networks (1) Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Data Mining 1 / 49 Do you like noodles? Do you like noodles? Race Gender Yes No Black Male 10
More informationReasoning Under Uncertainty: Belief Network Inference
Reasoning Under Uncertainty: Belief Network Inference CPSC 322 Uncertainty 5 Textbook 10.4 Reasoning Under Uncertainty: Belief Network Inference CPSC 322 Uncertainty 5, Slide 1 Lecture Overview 1 Recap
More informationDirected Graphical Models or Bayesian Networks
Directed Graphical Models or Bayesian Networks Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Bayesian Networks One of the most exciting recent advancements in statistical AI Compact
More informationTópicos Especiais em Modelagem e Análise - Aprendizado por Máquina CPS863
Tópicos Especiais em Modelagem e Análise - Aprendizado por Máquina CPS863 Daniel, Edmundo, Rosa Terceiro trimestre de 2012 UFRJ - COPPE Programa de Engenharia de Sistemas e Computação Bayesian Networks
More informationIntelligent Systems: Reasoning and Recognition. Reasoning with Bayesian Networks
Intelligent Systems: Reasoning and Recognition James L. Crowley ENSIMAG 2 / MoSIG M1 Second Semester 2016/2017 Lesson 13 24 march 2017 Reasoning with Bayesian Networks Naïve Bayesian Systems...2 Example
More informationConditional Independence
Conditional Independence Sargur Srihari srihari@cedar.buffalo.edu 1 Conditional Independence Topics 1. What is Conditional Independence? Factorization of probability distribution into marginals 2. Why
More informationCS839: Probabilistic Graphical Models. Lecture 2: Directed Graphical Models. Theo Rekatsinas
CS839: Probabilistic Graphical Models Lecture 2: Directed Graphical Models Theo Rekatsinas 1 Questions Questions? Waiting list Questions on other logistics 2 Section 1 1. Intro to Bayes Nets 3 Section
More informationLecture 5: Bayesian Network
Lecture 5: Bayesian Network Topics of this lecture What is a Bayesian network? A simple example Formal definition of BN A slightly difficult example Learning of BN An example of learning Important topics
More informationTDT70: Uncertainty in Artificial Intelligence. Chapter 1 and 2
TDT70: Uncertainty in Artificial Intelligence Chapter 1 and 2 Fundamentals of probability theory The sample space is the set of possible outcomes of an experiment. A subset of a sample space is called
More informationSTAT 598L Probabilistic Graphical Models. Instructor: Sergey Kirshner. Bayesian Networks
STAT 598L Probabilistic Graphical Models Instructor: Sergey Kirshner Bayesian Networks Representing Joint Probability Distributions 2 n -1 free parameters Reducing Number of Parameters: Conditional Independence
More informationMachine Learning Summer School
Machine Learning Summer School Lecture 1: Introduction to Graphical Models Zoubin Ghahramani zoubin@eng.cam.ac.uk http://learning.eng.cam.ac.uk/zoubin/ epartment of ngineering University of ambridge, UK
More informationCOMP538: Introduction to Bayesian Networks
COMP538: Introduction to Bayesian Networks Lecture 2: Bayesian Networks Nevin L. Zhang lzhang@cse.ust.hk Department of Computer Science and Engineering Hong Kong University of Science and Technology Fall
More information1 : Introduction. 1 Course Overview. 2 Notation. 3 Representing Multivariate Distributions : Probabilistic Graphical Models , Spring 2014
10-708: Probabilistic Graphical Models 10-708, Spring 2014 1 : Introduction Lecturer: Eric P. Xing Scribes: Daniel Silva and Calvin McCarter 1 Course Overview In this lecture we introduce the concept of
More informationIntroduction to Artificial Intelligence. Unit # 11
Introduction to Artificial Intelligence Unit # 11 1 Course Outline Overview of Artificial Intelligence State Space Representation Search Techniques Machine Learning Logic Probabilistic Reasoning/Bayesian
More informationTópicos Especiais em Modelagem e Análise - Aprendizado por Máquina CPS863
Tópicos Especiais em Modelagem e Análise - Aprendizado por Máquina CPS863 Daniel, Edmundo, Rosa Terceiro trimestre de 2012 UFRJ - COPPE Programa de Engenharia de Sistemas e Computação Motivação Grandes
More informationBayesian Networks. Machine Learning, Fall Slides based on material from the Russell and Norvig AI Book, Ch. 14
Bayesian Networks Machine Learning, Fall 2010 Slides based on material from the Russell and Norvig AI Book, Ch. 14 1 Administrativia Bayesian networks The inference problem: given a BN, how to make predictions
More informationA 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 information2 : Directed GMs: Bayesian Networks
10-708: Probabilistic Graphical Models, Spring 2015 2 : Directed GMs: Bayesian Networks Lecturer: Eric P. Xing Scribes: Yi Cheng, Cong Lu 1 Notation Here the notations used in this course are defined:
More informationBayesian Networks to design optimal experiments. Davide De March
Bayesian Networks to design optimal experiments Davide De March davidedemarch@gmail.com 1 Outline evolutionary experimental design in high-dimensional space and costly experimentation the microwell mixture
More informationChapter 16. Structured Probabilistic Models for Deep Learning
Peng et al.: Deep Learning and Practice 1 Chapter 16 Structured Probabilistic Models for Deep Learning Peng et al.: Deep Learning and Practice 2 Structured Probabilistic Models way of using graphs to describe
More informationSTA 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 informationUsing Graphs to Describe Model Structure. Sargur N. Srihari
Using Graphs to Describe Model Structure Sargur N. srihari@cedar.buffalo.edu 1 Topics in Structured PGMs for Deep Learning 0. Overview 1. Challenge of Unstructured Modeling 2. Using graphs to describe
More informationCausal Directed Acyclic Graphs
Causal Directed Acyclic Graphs Kosuke Imai Harvard University STAT186/GOV2002 CAUSAL INFERENCE Fall 2018 Kosuke Imai (Harvard) Causal DAGs Stat186/Gov2002 Fall 2018 1 / 15 Elements of DAGs (Pearl. 2000.
More informationBayesian networks. Independence. Bayesian networks. Markov conditions Inference. by enumeration rejection sampling Gibbs sampler
Bayesian networks Independence Bayesian networks Markov conditions Inference by enumeration rejection sampling Gibbs sampler Independence if P(A=a,B=a) = P(A=a)P(B=b) for all a and b, then we call A and
More informationLearning Semi-Markovian Causal Models using Experiments
Learning Semi-Markovian Causal Models using Experiments Stijn Meganck 1, Sam Maes 2, Philippe Leray 2 and Bernard Manderick 1 1 CoMo Vrije Universiteit Brussel Brussels, Belgium 2 LITIS INSA Rouen St.
More informationSupplementary material to Structure Learning of Linear Gaussian Structural Equation Models with Weak Edges
Supplementary material to Structure Learning of Linear Gaussian Structural Equation Models with Weak Edges 1 PRELIMINARIES Two vertices X i and X j are adjacent if there is an edge between them. A path
More informationArrowhead completeness from minimal conditional independencies
Arrowhead completeness from minimal conditional independencies Tom Claassen, Tom Heskes Radboud University Nijmegen The Netherlands {tomc,tomh}@cs.ru.nl Abstract We present two inference rules, based on
More informationMachine 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 information2 : Directed GMs: Bayesian Networks
10-708: Probabilistic Graphical Models 10-708, Spring 2017 2 : Directed GMs: Bayesian Networks Lecturer: Eric P. Xing Scribes: Jayanth Koushik, Hiroaki Hayashi, Christian Perez Topic: Directed GMs 1 Types
More informationBayesian Networks for Causal Analysis
Paper 2776-2018 Bayesian Networks for Causal Analysis Fei Wang and John Amrhein, McDougall Scientific Ltd. ABSTRACT Bayesian Networks (BN) are a type of graphical model that represent relationships between
More informationTowards an extension of the PC algorithm to local context-specific independencies detection
Towards an extension of the PC algorithm to local context-specific independencies detection Feb-09-2016 Outline Background: Bayesian Networks The PC algorithm Context-specific independence: from DAGs to
More informationIntroduction to Bayes Nets. CS 486/686: Introduction to Artificial Intelligence Fall 2013
Introduction to Bayes Nets CS 486/686: Introduction to Artificial Intelligence Fall 2013 1 Introduction Review probabilistic inference, independence and conditional independence Bayesian Networks - - What
More informationBayesian 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 informationParametrizations of Discrete Graphical Models
Parametrizations of Discrete Graphical Models Robin J. Evans www.stat.washington.edu/ rje42 10th August 2011 1/34 Outline 1 Introduction Graphical Models Acyclic Directed Mixed Graphs Two Problems 2 Ingenuous
More informationLearning Multivariate Regression Chain Graphs under Faithfulness
Sixth European Workshop on Probabilistic Graphical Models, Granada, Spain, 2012 Learning Multivariate Regression Chain Graphs under Faithfulness Dag Sonntag ADIT, IDA, Linköping University, Sweden dag.sonntag@liu.se
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationMarkov Independence (Continued)
Markov Independence (Continued) As an Undirected Graph Basic idea: Each variable V is represented as a vertex in an undirected graph G = (V(G), E(G)), with set of vertices V(G) and set of edges E(G) the
More informationNaïve Bayes Classifiers
Naïve Bayes Classifiers Example: PlayTennis (6.9.1) Given a new instance, e.g. (Outlook = sunny, Temperature = cool, Humidity = high, Wind = strong ), we want to compute the most likely hypothesis: v NB
More informationBayesian Networks. Semantics of Bayes Nets. Example (Binary valued Variables) CSC384: Intro to Artificial Intelligence Reasoning under Uncertainty-III
CSC384: Intro to Artificial Intelligence Reasoning under Uncertainty-III Bayesian Networks Announcements: Drop deadline is this Sunday Nov 5 th. All lecture notes needed for T3 posted (L13,,L17). T3 sample
More informationMachine Learning 4771
Machine Learning 4771 Instructor: Tony Jebara Topic 16 Undirected Graphs Undirected Separation Inferring Marginals & Conditionals Moralization Junction Trees Triangulation Undirected Graphs Separation
More informationIntroduction to Bayesian Networks. Probabilistic Models, Spring 2009 Petri Myllymäki, University of Helsinki 1
Introduction to Bayesian Networks Probabilistic Models, Spring 2009 Petri Myllymäki, University of Helsinki 1 On learning and inference Assume n binary random variables X1,...,X n A joint probability distribution
More informationBayesian Approaches Data Mining Selected Technique
Bayesian Approaches Data Mining Selected Technique Henry Xiao xiao@cs.queensu.ca School of Computing Queen s University Henry Xiao CISC 873 Data Mining p. 1/17 Probabilistic Bases Review the fundamentals
More informationStatistical 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 informationCPSC 540: Machine Learning
CPSC 540: Machine Learning Undirected Graphical Models Mark Schmidt University of British Columbia Winter 2016 Admin Assignment 3: 2 late days to hand it in today, Thursday is final day. Assignment 4:
More informationCognitive Systems 300: Probability and Causality (cont.)
Cognitive Systems 300: Probability and Causality (cont.) David Poole and Peter Danielson University of British Columbia Fall 2013 1 David Poole and Peter Danielson Cognitive Systems 300: Probability and
More informationProbabilistic Reasoning. (Mostly using Bayesian Networks)
Probabilistic Reasoning (Mostly using Bayesian Networks) Introduction: Why probabilistic reasoning? The world is not deterministic. (Usually because information is limited.) Ways of coping with uncertainty
More informationIntroduction to Bayesian Networks
Introduction to Bayesian Networks The two-variable case Assume two binary (Bernoulli distributed) variables A and B Two examples of the joint distribution P(A,B): B=1 B=0 P(A) A=1 0.08 0.02 0.10 A=0 0.72
More informationRepresentation 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 informationECE521 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 informationLecture 4 October 18th
Directed and undirected graphical models Fall 2017 Lecture 4 October 18th Lecturer: Guillaume Obozinski Scribe: In this lecture, we will assume that all random variables are discrete, to keep notations
More informationFaithfulness of Probability Distributions and Graphs
Journal of Machine Learning Research 18 (2017) 1-29 Submitted 5/17; Revised 11/17; Published 12/17 Faithfulness of Probability Distributions and Graphs Kayvan Sadeghi Statistical Laboratory University
More informationReview: Bayesian learning and inference
Review: Bayesian learning and inference Suppose the agent has to make decisions about the value of an unobserved query variable X based on the values of an observed evidence variable E Inference problem:
More informationLearning causal network structure from multiple (in)dependence models
Learning causal network structure from multiple (in)dependence models Tom Claassen Radboud University, Nijmegen tomc@cs.ru.nl Abstract Tom Heskes Radboud University, Nijmegen tomh@cs.ru.nl We tackle the
More informationCausal Reasoning with Ancestral Graphs
Journal of Machine Learning Research 9 (2008) 1437-1474 Submitted 6/07; Revised 2/08; Published 7/08 Causal Reasoning with Ancestral Graphs Jiji Zhang Division of the Humanities and Social Sciences California
More informationLearning from Sensor Data: Set II. Behnaam Aazhang J.S. Abercombie Professor Electrical and Computer Engineering Rice University
Learning from Sensor Data: Set II Behnaam Aazhang J.S. Abercombie Professor Electrical and Computer Engineering Rice University 1 6. Data Representation The approach for learning from data Probabilistic
More informationCausal Models. Macartan Humphreys. September 4, Abstract Notes for G8412. Some background on DAGs and questions on an argument.
Causal Models Macartan Humphreys September 4, 2018 Abstract Notes for G8412. Some background on DAGs and questions on an argument. 1 A note on DAGs DAGs directed acyclic graphs are diagrams used to represent
More informationSTATISTICAL METHODS IN AI/ML Vibhav Gogate The University of Texas at Dallas. Bayesian networks: Representation
STATISTICAL METHODS IN AI/ML Vibhav Gogate The University of Texas at Dallas Bayesian networks: Representation Motivation Explicit representation of the joint distribution is unmanageable Computationally:
More informationCausal Effect Identification in Alternative Acyclic Directed Mixed Graphs
Proceedings of Machine Learning Research vol 73:21-32, 2017 AMBN 2017 Causal Effect Identification in Alternative Acyclic Directed Mixed Graphs Jose M. Peña Linköping University Linköping (Sweden) jose.m.pena@liu.se
More informationProbabilistic Causal Models
Probabilistic Causal Models A Short Introduction Robin J. Evans www.stat.washington.edu/ rje42 ACMS Seminar, University of Washington 24th February 2011 1/26 Acknowledgements This work is joint with Thomas
More informationDecision-Theoretic Specification of Credal Networks: A Unified Language for Uncertain Modeling with Sets of Bayesian Networks
Decision-Theoretic Specification of Credal Networks: A Unified Language for Uncertain Modeling with Sets of Bayesian Networks Alessandro Antonucci Marco Zaffalon IDSIA, Istituto Dalle Molle di Studi sull
More information4.1 Notation and probability review
Directed and undirected graphical models Fall 2015 Lecture 4 October 21st Lecturer: Simon Lacoste-Julien Scribe: Jaime Roquero, JieYing Wu 4.1 Notation and probability review 4.1.1 Notations Let us recall
More informationPROBABILISTIC REASONING SYSTEMS
PROBABILISTIC REASONING SYSTEMS In which we explain how to build reasoning systems that use network models to reason with uncertainty according to the laws of probability theory. Outline Knowledge in uncertain
More informationProbabilistic Graphical Models and Bayesian Networks. Artificial Intelligence Bert Huang Virginia Tech
Probabilistic Graphical Models and Bayesian Networks Artificial Intelligence Bert Huang Virginia Tech Concept Map for Segment Probabilistic Graphical Models Probabilistic Time Series Models Particle Filters
More informationBayesian Networks aka belief networks, probabilistic networks. Bayesian Networks aka belief networks, probabilistic networks. An Example Bayes Net
Bayesian Networks aka belief networks, probabilistic networks A BN over variables {X 1, X 2,, X n } consists of: a DAG whose nodes are the variables a set of PTs (Pr(X i Parents(X i ) ) for each X i P(a)
More informationCausal Bayesian networks. Peter Antal
Causal Bayesian networks Peter Antal antal@mit.bme.hu A.I. 4/8/2015 1 Can we represent exactly (in)dependencies by a BN? From a causal model? Suff.&nec.? Can we interpret edges as causal relations with
More informationSummary of the Bayes Net Formalism. David Danks Institute for Human & Machine Cognition
Summary of the Bayes Net Formalism David Danks Institute for Human & Machine Cognition Bayesian Networks Two components: 1. Directed Acyclic Graph (DAG) G: There is a node for every variable D: Some nodes
More informationGraphical Models. Chapter Conditional Independence and Factor Models
Chapter 21 Graphical Models We have spent a lot of time looking at ways of figuring out how one variable (or set of variables) depends on another variable (or set of variables) this is the core idea in
More informationDirected acyclic graphs with edge-specific bounds
Biometrika (2012), 99,1,pp. 115 126 doi: 10.1093/biomet/asr059 C 2011 Biometrika Trust Advance Access publication 20 December 2011 Printed in Great Britain Directed acyclic graphs with edge-specific bounds
More informationArtificial Intelligence Bayes Nets: Independence
Artificial Intelligence Bayes Nets: Independence Instructors: David Suter and Qince Li Course Delivered @ Harbin Institute of Technology [Many slides adapted from those created by Dan Klein and Pieter
More informationLecture 17: May 29, 2002
EE596 Pat. Recog. II: Introduction to Graphical Models University of Washington Spring 2000 Dept. of Electrical Engineering Lecture 17: May 29, 2002 Lecturer: Jeff ilmes Scribe: Kurt Partridge, Salvador
More informationIntroduction to Probabilistic Graphical Models
Introduction to Probabilistic Graphical Models Kyu-Baek Hwang and Byoung-Tak Zhang Biointelligence Lab School of Computer Science and Engineering Seoul National University Seoul 151-742 Korea E-mail: kbhwang@bi.snu.ac.kr
More informationAutomatic Causal Discovery
Automatic Causal Discovery Richard Scheines Peter Spirtes, Clark Glymour Dept. of Philosophy & CALD Carnegie Mellon 1 Outline 1. Motivation 2. Representation 3. Discovery 4. Using Regression for Causal
More informationAxioms of Probability? Notation. Bayesian Networks. Bayesian Networks. Today we ll introduce Bayesian Networks.
Bayesian Networks Today we ll introduce Bayesian Networks. This material is covered in chapters 13 and 14. Chapter 13 gives basic background on probability and Chapter 14 talks about Bayesian Networks.
More informationBuilding Bayesian Networks. Lecture3: Building BN p.1
Building Bayesian Networks Lecture3: Building BN p.1 The focus today... Problem solving by Bayesian networks Designing Bayesian networks Qualitative part (structure) Quantitative part (probability assessment)
More informationDirected acyclic graphs and the use of linear mixed models
Directed acyclic graphs and the use of linear mixed models Siem H. Heisterkamp 1,2 1 Groningen Bioinformatics Centre, University of Groningen 2 Biostatistics and Research Decision Sciences (BARDS), MSD,
More informationPart I Qualitative Probabilistic Networks
Part I Qualitative Probabilistic Networks In which we study enhancements of the framework of qualitative probabilistic networks. Qualitative probabilistic networks allow for studying the reasoning behaviour
More informationIntroduction to Probabilistic Graphical Models
Introduction to Probabilistic Graphical Models Franz Pernkopf, Robert Peharz, Sebastian Tschiatschek Graz University of Technology, Laboratory of Signal Processing and Speech Communication Inffeldgasse
More informationGraphical 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 informationEquivalence in Non-Recursive Structural Equation Models
Equivalence in Non-Recursive Structural Equation Models Thomas Richardson 1 Philosophy Department, Carnegie-Mellon University Pittsburgh, P 15213, US thomas.richardson@andrew.cmu.edu Introduction In the
More informationBayesian Networks. Motivation
Bayesian Networks Computer Sciences 760 Spring 2014 http://pages.cs.wisc.edu/~dpage/cs760/ Motivation Assume we have five Boolean variables,,,, The joint probability is,,,, How many state configurations
More informationAbstract. Three Methods and Their Limitations. N-1 Experiments Suffice to Determine the Causal Relations Among N Variables
N-1 Experiments Suffice to Determine the Causal Relations Among N Variables Frederick Eberhardt Clark Glymour 1 Richard Scheines Carnegie Mellon University Abstract By combining experimental interventions
More information