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Chapter 15 Chapter 15 Probabilistic Reasoning over TimeProbabilistic Reasoning over Time
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Outline Outline
• Time and UncertaintyTime and Uncertainty
• Inference: Filtering, Prediction, SmoothingInference: Filtering, Prediction, Smoothing
• Hidden Markov modelsHidden Markov models
• Brief Introduction to Kalman FiltersBrief Introduction to Kalman Filters
• Dynamic Bayesian networksDynamic Bayesian networks
• Particle FilteringParticle Filtering
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Time and uncertaintyTime 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)Markov processes (Markov chains)
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ExampleExample
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Inference tasksInference tasks
tt
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FilteringFiltering
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Filtering exampleFiltering example
0
1 1 0 0
1 1 1 1 1
Day 1, U1=true
( ) ( | ) ( )
0.7,0.3 0.5 0.3,0.7 0.5 0.5,0.5
( | ) ( | ) ( ) 0.9,0.2 0.5,0.5
0.45,0.1 0.818,0.182
r
p R p R r p r
p R u p u R p R
Rt P(Ut)
t 0.9
f 0.2
Rt-1 P(Rt)
t 0.7
f 0.3
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Filtering exampleFiltering example
1
2 1 2 1 1 1
2 1 2 2 2 2 1
Day 2, U2=true
( | ) ( | ) ( | )
0.7,0.3 0.818 0.3,0.7 0.182 0.627,0.373
( |, , ) ( | ) ( | ) 0.9,0.2 0.627,0.373
0.565,0.075 0.883,0.117
r
p R u p R r p r u
p R u u p u R p R u
Rt-1 P(Rt)
t 0.7
f 0.3
Rt P(Ut)
t 0.9
f 0.2
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SmoothingSmoothing
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Smoothing exampleSmoothing example
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2 1 2 2 2 2 1(u | R ) ( | ) (| )p( | )
=0.9 1 0.7,0.3 0.2 1 0.3,0.7 0.69,0.41
r
p p u r p r r R
Rt-1 P(Rt)
t 0.7
f 0.3
Rt P(Ut)
t 0.9
f 0.2
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Most likely explanationMost likely explanation
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Viterbi exampleViterbi example
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Hidden Markov modelsHidden Markov models
:
: :
( | ) ( | ) ( | ) ( | )
( | )
1 1 1 1 1 1
1 1
tt t t t t t t tx
t t t
P X e P e X P X x P x e
f P x e
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Country dance algorithmCountry dance algorithm
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Country dance algorithmCountry dance algorithm
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Country dance algorithmCountry dance algorithm
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Country dance algorithmCountry dance algorithm
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Kalman FiltersKalman Filters
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Updating Gaussian distributionsUpdating Gaussian distributions
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Simple 1-D exampleSimple 1-D example
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General Kalman updateGeneral Kalman update
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2-D tracking example: Filtering2-D tracking example: Filtering
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2-D tracking example: smoothing2-D tracking example: smoothing
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Where it breaksWhere it breaks
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Dynamic Bayesian networksDynamic Bayesian networks
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DBNs vs. HMMsDBNs vs. HMMs
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DBNs vs Kalman FiltersDBNs vs Kalman Filters
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Exact inference in DBNsExact inference in DBNs
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Likelihood weighting for DBNsLikelihood weighting for DBNs
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Particle FilteringParticle Filtering
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Particle Filtering contd.Particle Filtering contd.
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Particle ltering performanceParticle ltering performance
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Chapter 15, Sections 1-5 Chapter 15, Sections 1-5 SummarySummary
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