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How is the Bellman equation of optimality calculated?
The Bellman equation of optimality Bellman proved that the optimal state value function in a state s is equal to the action a, which gives us the maximum possible expected immediate reward, plus the discounted long-term reward for the next state s’:
How to formulate Bellman equation for a given MDP?
So, this is how we can formulate Bellman Expectation Equation for a given MDP to find it’s State-Value Function and State-Action Value Function. But, it does not tell us the best way to behave in an MDP. For that let’s talk about what is meant by Optimal Value and Optimal Policy Function.
What is the Q function in Bellman equation?
In post 2 we extended the definition of state-value function to state-action pairs, defining a value for each state-action pair, which is called the action-value function, also known as Q-function or simply Q.
How is the Bellman equation simplified in dynamic programming?
Because r is governed by a Markov process, dynamic programming simplifies the problem significantly. Then the Bellman equation is simply: V ( a , r ) = max 0 ≤ c ≤ a { u ( c ) + β ∫ V ( ( 1 + r ) ( a − c ) , r ′ ) Q ( r , d μ r ) } .
How is the Bellman equation similar to the state value function?
The Bellman equation for the action-value function is similar in that it is a recursive equation for the value of a state-action pair of future possible pairs. Unlike the Bellman equation for the state-value function, however, we also want a recursive equation for the value of one state-action pair in terms of the next state-action pair.
How is Bellman expectation equation used in reinforcement learning?
First, let’s understand Bellman Expectation Equation for State-Value Function with the help of a backup diagram: This backup diagram describes the value of being in a particular state. From the state s there is some probability that we take both the actions. There is a Q-value (State-action value function) for each of the action.