A Tutorial on the Partially Observable Markov

Decision Process and Its Applications

Lawrence Carin

June 7,2006

Outline

Overview of Markov decision Processes

(MDPs)

Introduction to partially observable decision

processes (POMDPs)

Some applications of POMDPs

Conclusions

Overview of MDPs

Introduction to POMDPs model

Some applications of POMDPs

Conclusions

Markov decision processes

The MDP is defined by the tuple < S, A, T, R >

S is a finite set of states of the world.

A is a finite set of actions.

T: SA (S) is the state-transition function, the probability of an action changing the the world state from one to another,T(s, a, s’).

R: SA is the reward for the agent in a given world

state after performing an action, R(s, a).

Two properties of the MDP

• The action-dependent state transition is Markovian

• The state is fully observable after taking action aIllustration of MDPs

Markov decision processes

AGENT

WORLD: T(s,a, s’)Sta

te

s

Act

ion

a

as :

Markov decision processes

Objective of MDPs

• Finding the optimal policy , mapping state s to action a in order to maximize the value function V(s).

))'()',,(),((max)('

s

asVsasTasRsV

Overview of MDPs

Introduction to POMDPs

Some applications of POMDPs

Conclusions

S, A, T, and R are defined the same as in MDPs

is a finite set of observations the agent can experience its world.

O: SA () is the observation function, the probability of making a certain observation after performing a particular action, landing in state s’, O(s’, a, o).

The POMDP is defined by the tuple < S, A, T, R, , O >

Introduction to POMDPs

Introduction to POMDPs

Differences between MDPs and POMDPs• The state is hidden after taking action a.

• The hidden state information is inferred from the action-state dependent observation function O(s’, a, o).

Uncertainty of state s in POMDPs

Introduction to POMDPs

A new concept in POMDPs: Belief State b(s)b(st) = Pr(st = s | o1, a1, o2, a2, …, ot-1, at-1, ot)

Introduction to POMDPs

),|Pr(

)()',,(),,'( )'('

bao

sbsasToasOsb Ss

s1o1

o2b

b’=T(b|a, o1)

b’=T(b|a, o2)

n control interval remaining n-1 control interval remainings2 s2s3

s3

s1

(1)

The belief state b(s) evolves according to Bayes rule

Introduction to POMDPs

Illustration of POMDPs

SE:

AGENT

b

WORLD: T(s,a, s’)

O(s’, a, o)

Obse

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n

o

Act

ion

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SE: State Estimator using (1)

: Policy Search

Introduction to POMDPs

• Finding the optimal policy for POMDPs, mapping belief point b to action a in order to maximize the value function V(b).

))'((),'|Pr(),|'Pr()(),()(max

))'((){,'|Pr(),|'Pr()(max)(

1'

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oan

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n

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Expected immediate reward

Objective of POMDPs

(2)

][max])()([max)( bsbsbV knk

Ss

knkn

Pr(S1

)0 1

V(b

)

p1

p2

p3

p4

p5

a(p1) a(p2) a(p5)

• Piecewise linearity and convexity of optimal value function for finite horizon in POMDPs

Introduction to POMDPs

Optimal value function

(3)

Substituting (3), (1) into (2)

)'(),'|(),|'(),()(max

)'(),'|(),|'()(max),()(max

)'(),|Pr(),()(max)(

'

),,(1

'1

1

Ss o s

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Aa

o s s

kn

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on

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Maximizing to obtain the index l

-vector of belief point b

Optimal value of belief point b

(4)

Introduction to POMDPs

Introduction to POMDPs

Approaches to solving POMDPs problem• Exact algorithms: finding all -vectors for the

whole

belief space which is exact but intractable for large

size problems.

• Approximate algorithms: finding -vectors of a

subset of the belief space, which is fast and can deal

with large size problems.

Point-based value iteration (PBVI)

b5b1 b3 b4b0

• focus on a finite set of belief points

• maintain an -vector for each point

Point-Based Value Iteration

• RBVI maintains an -vector for each convex region over which the optimal value function is linear.

• RBVI simultaneously determines the -vectors for all relevant convex regions based on all available belief points.

Region-Based Value Iteration (RBVI)

The piecewise linear value function:

which can be reformulated as

by introducing hidden variables z(b)=k, denoting b Bk

RBVI (Contd)

The belief space is partitioned using hyper-ellipsoids,

Then we have

RBVI (Contd)

The joint distribution of V(b) and b can be written as

where

Expectation-Maximization (EM) Estimation:

RBVI (Contd)

E step:

M step:

Overview of MDPs

Introduction to POMDPs model

Some applications of POMDPs

Conclusions

Applications of POMDPs• Application of Partially Observable Markov Decision

Processes to robot navigation in a Minefield

• Application of Partially Observable Markov Decision

Processes to feature selection

• Application of Partially Observable Markov Decision

Processes to sensor scheduling

Applications of POMDPsSome considerations in applying

POMDPs to new problems

• How to define the state

• How to obtain the transition and observation matrix

• How to set the reward

References1. Leslie Pack Kaelbling, Michael L. Littman and Anthony R.

Cassandra. Planning and Acting in Partially Observable Stochastic Domains. Artificial Intelligence, Vol. 101,1998.

2. Smallwood, R. D., and Sondik, E. J. 1973. The optimal control of partially observable markov processes over a finite horizon. Operational Research 21:1071–1088.

3. J. Pineau, G. Gordon & S. Thrun. Point-based value iteration: An anytime algorithm for POMDPs. International Joint Conference on Artificial Intelligence (IJCAI), Acapulco, Mexico, Aug. 2003.

4. D. P. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific, Blemont, Massachusetts,2001, Vol.1 & Vol.2.

5. Bellman, R. 1957. Dynamic Programming. Princeton University Press.