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

Daphne Koller

Markov Assumption

Daphne Koller

Time Invariance• Template probability model P(X’ | X)• For all t:

Daphne Koller

Template Transition Model

Location’

Failure’

Obs’

Location

Failure

Time slice t Time slice t+1

Velocity Velocity’

Weather Weather’

Daphne Koller

Location0

Failure0

Time slice 0

Velocity0

Weather0

Obs0

Initial State Distribution

Daphne Koller

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

Daphne Koller

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

Daphne Koller

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

Daphne Koller

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

Daphne Koller

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