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INTRODUCTION TO GRAPHICAL MODELS SLIDE CREDITS: KEVIN MURPHY, MARK PASHKIN, ZOUBIN GHAHRAMANI AND JEFF BILMES CS188: Computational Models of Human Behavior

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Reasoning under uncertainty In many settings, we need to understand what is going on in a system when we have imperfect or incomplete information For example, we might deploy a burglar alarm to detect intruders – But the sensor could be triggered by other events, e.g., earth-quake Probabilities quantify the uncertainties regarding the occurrence of events

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Probability spaces A probability space represents our uncertainty regarding an experiment It has two parts: – A sample space, which is the set of outcomes – the probability measure P, which is a real function of the subsets of A set of outcomes A is called an event. P(A) represents how likely it is that the experiments actual outcome be a member of A

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An example If our experiment is to deploy a burglar alarm and see if it works, then there could be four outcomes: = {(alarm, intruder), (no alarm, intruder), (alarm, no intruder), (no alarm, no intruder)} Our choice of P has to obey these simple rules …

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The three axioms of probability theory P(A)0 for all events A P( )=1 P(A U B) = P(A) + P(B) for disjoint events A and B

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Some consequences of the axioms

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Example Lets assign a probability to each outcome ω These probabilities must be non-negative and sum to one intruderno intruder alarm no alarm

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Conditional Probability

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Marginal probability Marginal probability is then the unconditional probability P(A) of the event A; that is, the probability of A, regardless of whether event B did or did not occur. For example, if there are two possible outcomes corresponding to events B and B', this means that – P(A) = P(A B) + P(A B) This is called marginalization

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Example If P is defined by then P({(intruder, alarm)|(intruder, alarm),(no intruder, alarm)}) intruderno intruder alarm no alarm

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The product rule The probability that A and B both happen is the probability that A happens and B happens, given A has occurred

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The chain rule Applying the product rule repeatedly: P(A 1,A 2,…,A k ) = P(A 1 ) P(A 2 |A 1 )P(A 3 |A 2,A 1 )…P(A k |A k-1,…,A 1 ) Where P(A 3 |A 2,A 1 ) = P(A 3 |A 2 A 1 )

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Bayes rule Use the product rule both ways with P(A B) – P(A B) = P(A)P(B|A) – P(A B) = P(B)P(A|B)

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Random variables and densities

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Inference One of the central problems of computational probability theory Many problems can be formulated in these terms. Examples: – The probability that there is an intruder given the alarm went off is p I|A (true, true) Inference requires manipulating densities

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Probabilistic graphical models Combination of graph theory and probability theory – Graph structure specifies which parts of the system are directly dependent – Local functions at each node specify how different parts interaction Bayesian Networks = Probabilistic Graphical Models based on directed acyclic graph Markov Networks = Probabilistic Graphical Models based on undirected graph

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Some broad questions

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Bayesian Networks Nodes are random variables Edges represent dependence – no directed cycles allowed) P(X 1:N ) = P(X 1 )P(X 2 |X 1 )P(X 3 |X 1,X 2 ) = P(X i |X 1:i-1 ) = P(X i |X i ) x1x1 x2x2 x3x3 x5x5 x4x4 x7x7 x6x6

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Example Water sprinkler Bayes net P(C,S,R,W)=P(C)P(S|C)P(R|C,S)P(W|C,S,R) chain rule =P(C)P(S|C)P(R|C)P(W|C,S,R) since R S|C =P(C)P(S|C)P(R|C)P(W|S,R) since W C|R,S

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Inference

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Naïve inference

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Problem with naïve representation of the joint probability Problems with the working with the joint probability – Representation: big table of numbers is hard to understand – Inference: computing a marginal P(X i ) takes O(2 N ) time – Learning: there are O(2 N ) parameters to estimate Graphical models solve the above problems by providing a structured representation for the joint Graphs encode conditional independence properties and represent families of probability distribution that satisfy these properties

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Bayesian networks provide a compact representation of the joint probability

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Conditional probabilities

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Another example: medical diagnosis (classification)

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Approach: build a Bayes net and use Bayess rule to get class probability

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A very simple Bayes net: Naïve Bayes

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Naïve Bayes classifier for medical diagnosis

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Another commonly used Bayes net: Hidden Markov Model (HMM)

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Conditional independence properties of Bayesian networks: chains

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Conditional independence properties of Bayesian networks: common cause

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Conditional independence properties of Bayesian networks: explaining away

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Global Markov properties of DAGs

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Bayes ball algorithm

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Example

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Undirected graphical models

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Parameterization

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Clique potentials

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Interpretation of clique potentials

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Examples

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Joint distribution of an undirected graphical model Complexity scales exponentially as 2 n for binary random variable if we use a naïve approach to computing the partition function

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Max clique vs. sub-clique

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Log-linear models

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Summary

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From directed to undirected graphs

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Example of moralization

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Comparing directed and undirected models

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Expressive power xy w z xy z

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Coming back to inference

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Belief propagation in trees

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Learning

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Parameter Estimation

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Maximum-likelihood Estimation (MLE)

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Example: 1-D Gaussian

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MLE for Bayes Net

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MLE for Bayes Net with Discrete Nodes

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Parameter Estimation with Hidden Nodes Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z

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Why is learning harder?

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Where do hidden variables come from?

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Parameter Estimation with Hidden Nodes z z

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EM

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Different Learning Conditions StructureObservability FullPartial KnownClosed form searchEM UnknownLocal searchStructural EM

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