In the context of Reinforcement Learning, Partially Observable Markov Decision P

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In the context of Reinforcement Learning, Partially Observable Markov Decision Processes (POMDPs) extend MDPs to scenarios where the agent does not have full observability of the system state. This is particularly relevant in real-world applications where sensor noise, occlusions, or limited field of view prevent complete knowledge of the environment.
Given a POMDP defined by the tuple (S,A,T,R,Ω,O,γ)(S, A, T, R, Omega, O, gamma)(S,A,T,R,Ω,O,γ), where:
SSS is a finite set of states,
AAA is a finite set of actions,
T:S×A×S→[0,1]T: S times A times S to [0,1]T:S×A×S→[0,1] is the state transition probability function,
R:S×A→RR: S times A to mathbb{R}R:S×A→R is the reward function,
ΩOmegaΩ is a finite set of observations,
O:S×A×Ω→[0,1]O: S times A times Omega to [0,1]O:S×A×Ω→[0,1] is the observation probability function,
γ∈[0,1)gamma in [0,1)γ∈[0,1) is the discount factor.
Design an optimal policy π:B→Api: B to Aπ:B→A for a POMDP where BBB represents the belief state (a probability distribution over states). The optimal policy should maximize the expected sum of discounted rewards.
Tasks:
Formulate the Problem:Derive the belief update equation for the POMDP.
Represent the value function V(b)V(b)V(b) for belief states b∈Bb in Bb∈B.
Derive the Bellman Equation:Extend the Bellman equation to the belief space.
Algorithm Development:Propose a solution algorithm (e.g., Point-Based Value Iteration, PBVI) to approximate the optimal policy.
Provide pseudocode for the proposed algorithm.
Implementation:Implement the proposed algorithm in a programming language of your choice (Python is preferred).
Test your implementation on a benchmark POMDP problem (e.g., the Tiger problem).
Evaluation:Analyze the performance of your algorithm in terms of computational complexity and convergence.
Compare your results with other standard algorithms for solving POMDPs.

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