Probabilistic Graphical Models ALIREZA NOURIAN 

Meanings  Model: declarative representation of our understanding of the world  Experts knowledge or Learning  Probabilistic: handling uncertainty  Partial, noisy perceptions  Inherently stochastic phenomena  Graphical: relations between random variables  Bayesian Network: directed  Markov Network: undirected 2 

Real Models Bayesian Network Markov Network 3 

Image Segmentation 4 

Natural Language Processing  Named Entity Recognition  Joint Tagging 5 

Overview  Representation  Inference  Learning 6 

Bayesian Networks  Knowledge Engineering  Easy to design and maintain  e.g. Fault Diagnosis 7 

Bayesian Networks – Reasoning Causal Reasoning Evidential Reasoning 8 

Naïve Bayes Classifier  Strong Independence Assumption  Xi ⊥ Xj |C (i, j ∈ 1..n)  , 1 , … , = () =1 9 

Plate Models  Repeated Structure for Multiple Objects of the Same Type  Grade of student in course  Unrolled for 2 students and 2 courses 10 

Dynamic Bayesian Networks  Representation of Structured Distributions over Time  Markov Assumption  0.. = 0 =0 −1 +1  Time Invariance Assumption  +1 = ′ 11 

Vehicle Model (Dynamic Bayesian Network) 12 

Hidden Markov Models  Sort of DBN  State machine for State values  e.g. Speech Recognition 13 

Markov Networks  Conditional Random Fields  Log-Linear Models  Metric Markov Random Field  Image Segmentation  Close pixels have same label  Image Denoising  Close pixels have same color  Stereo Reconstruction  Close points have same depth 14 

Markov Network – Example  Named Entity Recognition  Features: word capitalized, word in atlas or name list, previous word is “Mrs”, next word is “Times”, … 15 

Inference  Conditional Probability Queries  = =  e.g. Fault Diagnosis  NP-Hard problem  Algorithms  Variable Elimination  Belief Propagation  Random Sampling  Maximum a Posteriori (MAP)  argmax = =  e.g. most likely Image Segmentation  NP-Hard problem  Algorithms  Variable Elimination  Belief Propagation  Optimization 16 

Sample MAP – Correspondence  Model  3D Reconstruction  Different images  Human Pose Detection  Different scanned meshes 17 

Inference – Variable Elimination  Reduce factors by Evidence  Eliminate non-query variables  Multiply all remaining factors  Renormalize 18 

Variable Elimination – Induced Graph 19 

Variable Elimination Ordering  Greedy Search  Heuristic Cost Function  min-neighbors: # neighbors in current graph  min-fill: number of new fill edges  … 20 

Robot Localization – Markov Random Field 21 

Robot Localization – Eliminate Poses then Landmarks 22 

Robot Localization – Eliminate Landmarks then Poses 23 

Robot Localization – Min-Fill Elimination 24 

Inference – Belief Propagation  Adjacent variable clusters pass information to each other 25 

Inference – Sampling  Estimating probability distribution from it’s samples  Minimum number of samples for estimation  Forward sampling (Bayesian Networks)  Sample variable given it’s parents  Markov Chain Monte Carlo (Markov Networks)  Sample current state given previous one  Gibbs Sampling  Sample one variable given others 26 

Inference – Decision Making 27  Influence diagram  Bayesian network with action and utility nodes 

Inference – Belief State Tracking  Inference in Temporal Model  e.g. Robot Localization 28 

Robot Localization – Belief State Tracking 29 

Learning Model from Data  Parameter Estimation  Maximum Likelihood Estimation  Structure Learning  Optimization over Structures  Likelihood of Data given Structure  Local Search in Structure Space 30 

Structure Learning - Local Search 31 

Summary  Representation  Directed and Undirected  Temporal and Plate Models  Inference  Exact and Approximate  Decision Making  Learning  Parameters and Structure  With and Without Complete Data 32 

Reference  Probabilistic Graphical Models by Daphne Koller  https://www.coursera.org/course/pgm 33