introduction to graphical models slide credits: kevin murphy, mark pashkin, zoubin ghahramani and...
TRANSCRIPT
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INTRODUCTION TO GRAPHICAL MODELSSLIDE 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 experiment’s 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
• Let’s assign a probability to each outcome ω
• These probabilities must be non-negative and sum to one
intruder no intruder
alarm 0.002 0.003
no alarm 0.001 0.994
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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(AB) + P(AB’)
• This is called marginalization
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Example• If P is defined by
then P({(intruder, alarm)|(intruder, alarm),(no intruder, alarm)})
intruder no intruder
alarm 0.002 0.003
no alarm 0.001 0.994
P({(intruder,alarm)} {(intruder,alarm),(no intruder,alarm)})({(intruder,alarm),(no intruder,alarm)})
P({(intruder,alarm)})({(intruder,alarm),(no intruder,alarm)})
0.0020.4
(0.002 0.003)
P
P
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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(A1,A2,…,Ak) = P(A1) P(A2|A1)P(A3|A2,A1)…P(Ak|Ak-1,…,A1)
• Where P(A3|A2,A1) = P(A3|A2A1)
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Bayes’ rule
• Use the product rule both ways with P(AB)– 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 pI|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(X1:N) = P(X1)P(X2|X1)P(X3|X1,X2) = P(Xi|X1:i-1) = P(Xi|Xi)
x1
x2
x3
x5x4
x7x6
1
N
i
1
N
i
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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(Xi) takes O(2N) time
– Learning: there are O(2N) 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 Bayes’s 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 2n 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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Log-linear models
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Log-linear models
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Summary
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Summary
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From directed to undirected graphs
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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
x y
w
z
x y
z
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Coming back to inference
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Coming back to inference
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Belief propagation in trees
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Belief propagation in trees
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Belief propagation in trees
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Belief propagation in trees
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Belief propagation in trees
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Belief propagation in trees
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Belief propagation in trees
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Belief propagation in trees
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Learning
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Parameter Estimation
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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
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MLE for Bayes’ Net with Discrete Nodes
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Parameter Estimation with Hidden Nodes
Z1 Z2 Z3 Z4 Z5 Z6
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
zz
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EM
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Different Learning Conditions
Structure Observability
Full Partial
Known Closed form search EM
Unknown Local search Structural EM