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CSE 473/573 Computer Vision and Image Processing (CVIP) Ifeoma Nwogu Lecture 27 – Overview of probability concepts 1

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Schedule Last class – Object recognition and bag-of-words Today – Overview of probability models Readings for today: – Lecture notes 2

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Random variables A random variable x denotes a quantity that is uncertain May be result of experiment (flipping a coin) or a real world measurements (measuring temperature) If observe several instances of x we get different values Some values occur more than others and this information is captured by a probability distribution 3Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Discrete Random Variables 4Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Continuous Random Variable 5Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Joint Probability Consider two random variables x and y If we observe multiple paired instances, then some combinations of outcomes are more likely than others This is captured in the joint probability distribution Written as Pr( x, y ) Can read Pr( x, y ) as “probability of x and – where x and y are scalars 6Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Joint Probability (2) We can definitely have more than two random variables and we would write Pr( x, y,z ) We may also write Pr( x ) - the joint probability of all of the elements of the multidimensional variable x = [x 1,x 2 ….x n ] and if y = [y 1,y 2 ….y n ] then we can write Pr( x, y ) to represent the joint distribution of all of the elements from multidimensional variables x and y. 7Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Joint Probability (3) 8Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 2D Hinton diagramJoint distributions of one discrete and one continuous variable

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Marginalization We can recover probability distribution of any variable in a joint distribution by integrating (or summing) over the other variables 9Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Marginalization We can recover probability distribution of any variable in a joint distribution by integrating (or summing) over the other variables 10Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Scale and Normalization

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Marginalization We can recover probability distribution of any variable in a joint distribution by integrating (or summing) over the other variables 11Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Marginalization We can recover probability distribution of any variable in a joint distribution by integrating (or summing) over the other variables Works in higher dimensions as well – leaves joint distribution between whatever variables are left 12Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Conditional Probability Conditional probability of x given that y = y 1 is relative propensity of variable x to take different outcomes given that y is fixed to be equal to y 1. Written as Pr( x | y=y 1 ) 13 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Conditional Probability Conditional probability can be extracted from joint probability Extract appropriate slice and normalize 14Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Conditional Probability More usually written in compact form Can be re-arranged to give 15Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Conditional Probability This idea can be extended to more than two variables 16Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Bayes’ Rule From before: Combining: Re-arranging: 17Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Bayes’ Rule Terminology Posterior – what we know about y after seeing x Prior – what we know about y before seeing x Likelihood – propensity for observing a certain value of x given a certain value of y Evidence –a constant to ensure that the left hand side is a valid distribution 18Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Independence If two variables x and y are independent then variable x tells us nothing about variable y (and vice-versa) 19Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Independence If two variables x and y are independent then variable x tells us nothing about variable y (and vice-versa) 20Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Independence When variables are independent, the joint factorizes into a product of the marginals: 21Computer vision: models, learning and inference. ©2011 Simon J.D. Prince

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Probabilistic graphical models 22

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What is a graphical model ? A graphical model is a way of representing probabilistic relationships between random variables. Variables are represented by nodes: Conditional (in)dependencies are represented by (missing) edges: Undirected edges simply give correlations between variables (Markov Random Field or Undirected Graphical model): Directed edges give causality relationships (Bayesian Network or Directed Graphical Model):

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Graphical Models e.g. Bayesian networks, Bayes nets, Belief nets, Markov networks, HMMs, Dynamic Bayes nets, etc. Themes: – representation – reasoning – learning Original PGM reference book by Nir Friedman and Daphne Koller.

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Motivation for PGMs Let X 1,…,X p be discrete random variables Let P be a joint distribution over X 1,…,X p If the variables are binary, then we need O(2 p ) parameters to describe P Can we do better? Key idea: use properties of independence

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Motivation for PGMs (2) Two variables X and Y are independent if – P(X = x|Y = y) = P(X = x) for all values x, y – That is, learning the values of Y does not change prediction of X If X and Y are independent then – P(X,Y) = P(X|Y)P(Y) = P(X)P(Y) In general, if X 1,…,X p are independent, then – P(X 1,…,X p )= P(X 1 )...P(X p ) – Requires O(n) parameters

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Motivation for PGMs (3) Unfortunately, most of random variables of interest are not independent of each other A more suitable notion is that of conditional independence Two variables X and Y are conditionally independent given Z if – P(X = x|Y = y,Z=z) = P(X = x|Z=z) for all values x,y,z – That is, learning the values of Y does not change prediction of X once we know the value of Z

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Probability theory provides the glue whereby the parts are combined, ensuring that the system as a whole is consistent, and providing ways to interface models to data. The graph theoretic side of graphical models provides both an intuitively appealing interface by which humans can model highly- interacting sets of variables as well as a data structure that lends itself naturally to the design of efficient general-purpose algorithms. Many of the classical multivariate probabilistic systems studied in fields such as statistics, systems engineering, information theory, pattern recognition and statistical mechanics are special cases of the general graphical model formalism -- examples include mixture models, factor analysis, hidden Markov models, and Kalman filters. In summary….

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Slide Credits Simon Prince – Computer vision, models and learning 29

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Next class More on Graphical models Readings for next lecture: – Lecture notes to be posted Readings for today: – Lecture notes to be posted 30

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Questions 31

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