daphne koller template models temporal models probabilistic graphical models representation
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Daphne Koller
Template Models
TemporalModels
ProbabilisticGraphicalModels
Representation
Daphne Koller
Distributions over Trajectories
• Pick time granularity • X(t) – variable X at time t• X(t:t’) = {X(t), …, X(t’)} (t t’)
• Want to represent P(X(t:t’)) for any t, t’
0 1 2 3 4 5
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Markov Assumption
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Time Invariance• Template probability model P(X’ | X)• For all t:
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Template Transition Model
Location’
Failure’
Obs’
Location
Failure
Time slice t Time slice t+1
Velocity Velocity’
Weather Weather’
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Location0
Failure0
Time slice 0
Velocity0
Weather0
Obs0
Initial State Distribution
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Location0
Failure0
Time slice 0
Velocity0
Weather0
Location1
Failure1
Obs1
Time slice 1
Velocity1
Weather1
Location2
Failure2
Obs2
Velocity2
Weather2
Time slice 2
Obs0
Ground Bayesian Network
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2-time-slice Bayesian Network
• A transition model (2TBN) over X1,…,Xn is specified as a BN fragment such that:– The nodes include X1’,…,Xn’ and a subset of X1,…,Xn
– Only the nodes X1’,…,Xn’ have parents and a CPD
• The 2TBN defines a conditional distribution
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Dynamic Bayesian Network• A dynamic Bayesian network (DBN)
over X1,…,Xn is defined by a– 2TBN BN over X1,…,Xn
– a Bayesian network BN(0) over X1(0)
,…,Xn
(0)
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Ground Network• For a trajectory over 0,…,T we define
a ground (unrolled network) such that– The dependency model for X1
(0) ,…,Xn
(0) is copied from BN(0)
– The dependency model for X1(t)
,…,Xn(t)
for all t > 0 is copied from BN
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S’
O’
S
Hidden Markov Models
S1
O1
S0 S2
O2
S3
O3
s1 s2 s3 s4
0.5
0.50.7
0.3
0.4
0.6
0.1
0.9
Daphne Koller
Consider a smoke detection tracking application, where we have 3 rooms connected in a row. Each room has a true smoke level (X) and a smoke level (Y) measured by a smoke detector situated in the middle of the room. Which of the following is the best DBN structure for this problem?
X’1
Y’1
X1
X’2
Y’2
X2
X’3
Y’3
X3
X’1
Y’1
X1
X’2
Y’2
X2
X’3
Y’3
X3
X’1
Y’1
X1
X’2
Y’2
X2
X’3
Y’3
X3
X’1
Y’1
X1
X’2
Y’2
X2
X’3
Y’3
X3
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Robot Localization
x
z
u
x
z
u
2
2
x
z
u
... t
t
x 1
1
0
10 t-1control signal
robot pose
sensor observation
map
Daphne KollerTim Huang, Dieter Koller, Jitendra Malik, Gary Ogasawara, Bobby Rao, Stuart Russell, J. Weber
Daphne Koller
Summary• DBNS are a compact representation for
encoding structured distributions over arbitrarily long temporal trajectories
• They make assumptions that may require appropriate model (re)design:–Markov assumption– Time invariance