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Cognitive Computer Vision Kingsley Sage [email protected] and Hilary Buxton [email protected] Prepared under ECVision Specific Action 8-3 http://www.ecvision.org Lecture 3 Graphical models Probabilistic graphical models – – – – Directed graphs Unidirected graphs Notation Rolling out over time What are graphical models? Represent salient relationships graphically e.g. What are probabilistic graphical models? A probabilistic graphical model is a type of probabilistic network that has roots in AI, statistics and neural networks Provides a clean mathematical formalism that makes it possible to understand the relationships between a wide variety of network based approaches to computation Allows to see different methods as instances of a broader probabilistic framework What are probabilistic graphical models? Probabilistic graphical models use graphs to represent and manipulate joint probability distributions Graphs can be directed – usually referred to as a belief network or Bayesian network Graphs can be undirected – usually referred to as a Markov Random Field A basis for algorithms for computation Joint probability – a reminder A probability dependent on more than one variable e.g. p(AND|a,b): a a b p(AND|a,b) 0 0 0 0 1 0 1 0 0 1 1 1 b Discrete case of a logic AND gate A continuous case where light values are high p(AND|a,b) Directed graphs A B Intuitively, the notion of causality (although this can be a philosophical argument) A B, so the value of A directly determines the value of B P(A,B) = P(B|A).P(A) Traffic lights model from lecture 2 as a directed graph OBSERVABLE HIDDEN hidden STOP GET READY TO STOP GET READY TO GO GO observable Examples of directed graphs Hidden Markov Models (later in the course) Kalman filters Factor analysis Independent component analysis Mixtures of Gaussians (later in the course) Probabilistic expert systems The list goes on … Joint probability – conditional independence N variables are conditionally independent if one value does not depend on the other e.g: A B Here, A and B are conditionally independent: A B iff P( A, B) P( A).P( B) But A and C and B and C are not: C P( A, B, C ) P(C | A, B) P(C ) P(C | A, B ).P( A, B ) P(C ) P(C | A, B ).P( A).P( B) Undirected graphs A B Intuitively, the notion of correlation (although this can be a philosophical argument) A B, so the values of A and B are interdependent Directed graphs can be converted into undirected graphs (but beyond the scope of this course) A undirected graph for a computer vision task Notation A B C Squares denote discrete nodes Circles denote continuous valued nodes Clear denotes hidden node Shaded denotes observed node Rolling out over time Probabilistic graphical model notation is very good at showing how models are propagated in time Expose the dependencies between the different elements of the graphical structure Rolling out our traffic light example over 2 time steps … OBSERVABLE t=1 t=2 hidden hidden observable observable HIDDEN STOP GET READY TO STOP GET READY TO GO GO Remember the concept of the temporal order of a model ? t=1 t=2 hidden hidden observable observable In this model, the value of the hidden nodes (and thus the observable ones) at time t+1 only depends on the previous time step t So this is a first order temporal model Remember the concept of the temporal order of a model ? t=1 t=2 t=3 t=4 hidden hidden hidden hidden observable observable observable observable A second order temporal model … … So why are graphical models relevant to Cognitive CV? Precisely because they allows us to see different methods as instances of a broader probabilistic framework These methods are the basis for our model of perception guided by expectation We can put our model of expectation on a solid theoretical foundation We can develop well-founded methods of learning rather than just being stuck with hand-coded models Summary Probabilistic graphical models put the formalisms on a well-founded mathematical basis We can distinguish directed and undirected graphs Here we concentrate on directed graphs that we can roll out over time easily Next time … A family of graphical models A lot of excellent reference material can be found at: http://cosco.hiit.fi/Teaching/GraphicalModels/Fall2003/material.html