Math Problem Statement
In a simple MDP, an agent is in a state s, and the actions it can take can lead to the following outcomes: • With probability 0.4, the agent transitions to state , with reward , and 5 IIT KHARAGPUR AI4ICPS I HUB FOUNDATION Hands-on Approach to AI, Cohort-2, July – October 2024 Assignment 7: Reinforcement Learning ϵ s′ R = 10 v(s′) = • With probability 0.6, the agent transitions to state , with reward , and 3. The discount factor is 0.5. Using Bellman equation, find the expected value of state . s′′ R = 2 v(s′) = γ s
Solution
Based on the image you provided, it seems that you're dealing with a Markov Decision Process (MDP) problem where you need to calculate the expected value of state using the Bellman equation. Let's break down the problem step by step.
Problem Details:
- Transition Probabilities:
- With probability , the agent transitions to state with a reward .
- With probability , the agent transitions to state with a reward .
- Discount Factor (): .
- Values of and : These values are not explicitly given in the problem. Therefore, let's assume them as and for now.
Bellman Equation:
The Bellman equation for the state-value function is given by: [ v(s) = \mathbb{E}[G_t \mid S_t = s] = \mathbb{E}[R_{t+1} + \gamma v(S_{t+1}) \mid S_t = s] ] In this context, the expected value of can be calculated as: Substituting the values:
Step-by-Step Calculation:
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Substitute and :
- If you have specific values for and , substitute them here.
But if these values are not given, let's assume:
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Expression:
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Simplify the Expression:
Conclusion:
- The final expression for is:
- To get a numerical value for , you would need the specific values for and . If those are not provided, this is the most simplified expression based on the given data.
Would you like more details or have any further questions?
Related Questions:
- What is the interpretation of the discount factor () in reinforcement learning?
- How does the Bellman equation help in solving MDP problems?
- What assumptions are made when calculating expected value in MDPs?
- Can you explain how the transition probabilities affect the state-value function?
- How would changing the reward values impact the value of ?
Tip:
When working with MDPs, always ensure that you have the correct and complete values for all states and rewards to accurately compute the expected state value.
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Math Problem Analysis
Mathematical Concepts
Markov Decision Process (MDP)
Bellman Equation
Expected Value Calculation
Formulas
Bellman equation for state-value function
Theorems
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Suitable Grade Level
Advanced undergraduate level
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