Transcript Ch15-Inference-Time

Chapter 15
Probabilistic Reasoning over Time
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Outline
• Time and Uncertainty
• Inference: Filtering, Prediction, Smoothing
• Hidden Markov models
• Brief Introduction to Kalman Filters
• Dynamic Bayesian networks
• Particle Filtering
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Time and uncertainty
• The world changes; we need to track and predict it
• Diabetes management vs vehicle diagnosis
• Basic idea: copy state and evidence variables for each time step
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Markov processes (Markov chains)
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Example
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Inference tasks
t
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Filtering
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Filtering example
Rt-1
P(Rt)
Rt
P(Ut)
t
0.7
t
0.9
f
0.3
f
0.2
Day 1, U1=true
p( R1 )   p( R1 | r0 ) p( r0 )
r0
 0.7,0.3  0.5  0.3,0.7  0.5  0.5,0.5 
p( R1 | u1 )   p(u1 | R1 ) p( R1 )    0.9,0.2    0.5,0.5 
   0.45,0.1  0.818,0.182 
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Filtering example
Rt-1
P(Rt)
t
0.7
f
0.3
Day 2, U2=true
p( R2 | u1 )   p( R2 | r1 ) p( r1 | u1 )
r1
 0.7,0.3  0.818  0.3,0.7  0.182  0.627,0.373 
p( R2 |, u1 , u2 )   p(u2 | R2 ) p( R2 | u1 )    0.9,0.2    0.627,0.373 
   0.565,0.075  0.883,0.117 
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Rt
P(Ut)
t
0.9
f
0.2
Smoothing
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Smoothing example
Rt-1
P(Rt)
t
0.7
f
0.3
Rt
P(Ut)
t
0.9
f
0.2
p(u 2 | R1 )   p(u2 | r2 ) p(| r2 )p( r2 | R1 ) 
r2
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=0.9  1  0.7,0.3  0.2  1  0.3,0.7  0.69,0.41 
Most likely explanation
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Viterbi example
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Hidden Markov models
P( X t 1 | et 1 )   P(et 1 | X t 1 ) x P( X t 1 | xt ) P( xt | e1:t )
t
f1:t  P( xt | e1:t )
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Country dance algorithm
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Country dance algorithm
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Country dance algorithm
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Country dance algorithm
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Kalman Filters
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Updating Gaussian distributions
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Simple 1-D example
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General Kalman update
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2-D tracking example: Filtering
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2-D tracking example: smoothing
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Where it breaks
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Dynamic Bayesian networks
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DBNs vs. HMMs
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DBNs vs Kalman Filters
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Exact inference in DBNs
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Likelihood weighting for DBNs
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Particle Filtering
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Particle Filtering contd.
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Particle ltering performance
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Chapter 15, Sections 1-5
Summary
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