1. A Tutorial on the Partially Observable Markov Decision Process and Its Applications Lawrence Carin June 7,2006

2. Outline  Overview of Markov decision Processes (MDPs)  Introduction to partially observable decision processes (POMDPs)  Some applications of POMDPs  Conclusions

3.  Overview of MDPs  Introduction to POMDPs model  Some applications of POMDPs  Conclusions

4. Markov decision processes The MDP is defined by the tuple  S is a finite set of states of the world.  A is a finite set of actions.  T : S  A   (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: S  A   is the reward for the agent in a given world state after performing an action, R(s, a).

5. Two properties of the MDP The action-dependent state transition is Markovian The state is fully observable after taking action a Illustration of MDPs Markov decision processes AGENT WORLD: T ( s,a, s’ ) State s Action a

6. 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).

7.  Overview of MDPs  Introduction to POMDPs  Some applications of POMDPs  Conclusions

8.  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 : S  A   (  ) 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 Introduction to POMDPs

9. 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

10. Introduction to POMDPs A new concept in POMDPs: Belief State b(s) b(s t ) = Pr(s t = s | o 1, a 1, o 2, a 2, …, o t-1, a t-1, o t )

11. Introduction to POMDPs s1s1 o1o1 o2o2 b b’=T(b|a, o 1 ) b’=T(b|a, o 2 ) n control interval remaining n-1 control interval remaining s2s2 s2s2 s3s3 s3s3 s1s1 (1) The belief state b(s) evolves according to Bayes rule

12. Introduction to POMDPs Illustration of POMDPs SE: AGENT b WORLD: T ( s,a, s’ ) O(s’, a, o) Observation o Action a SE: State Estimator using (1)  : Policy Search

13. 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). Expected immediate reward Objective of POMDPs (2)

14. Pr(S 1 ) 0 1 V(b)V(b) p1p1 p2p2 p3p3 p4p4 p5p5 a(p1)a(p1)a(p2)a(p2)a(p5)a(p5) Piecewise linearity and convexity of optimal value function for finite horizon in POMDPs Introduction to POMDPs Optimal value function (3)

15. Substituting (3), (1) into (2) Maximizing to obtain the index l  -vector of belief point b Optimal value of belief point b (4) Introduction to POMDPs

16. 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.

17. Point-based value iteration (PBVI) b5b5 b1b1 b3b3 b4b4 b0b0 focus on a finite set of belief points maintain an  -vector for each point Point-Based Value Iteration

18. 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)

19. The piecewise linear value function: which can be reformulated as by introducing hidden variables z(b) = k, denoting b  B k RBVI (Contd)

20. The belief space is partitioned using hyper-ellipsoids, Then we have RBVI (Contd)

21. The joint distribution of V(b) and b can be written as where Expectation-Maximization (EM) Estimation: RBVI (Contd) E step: M step:

22.  Overview of MDPs  Introduction to POMDPs model  Some applications of POMDPs  Conclusions

23. 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

24. Applications of POMDPs Some 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

25. References 1.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.