Markov Decision Processes

Markov Decision Processes (MDPs) are a mathematical framework used to model decision-making scenarios. In an MDP, an agent interacts with an environment by taking actions, and the environment responds with rewards and a new state. The goal of the agent is to maximize the total reward it receives over time. The agent must make decisions based on incomplete information and uncertainty about the effects of its actions.

Key Components

• A set of states
• A set of actions
• A transition function that describes the probability of moving from one state to another after taking an action
• A reward function that assigns a numeric reward to each state-action pair
• A discount factor that determines how much weight to give to future rewards

MDPs are often represented as a directed graph, where each node represents a state, and each edge represents an action. The transition function and reward function are associated with each edge. The discount factor is a parameter that is typically set between 0 and 1, where a higher value places more weight on future rewards.

A common algorithm for solving MDPs is the Bellman equation. The Bellman equation is a recursive equation that describes the optimal value function for each state. The value function represents the expected total reward that an agent will receive from a given state, taking into account the discount factor and the probabilities of transitioning to other states.

An important concept in MDPs is the policy. A policy is a mapping from states to actions that determines the agent's behavior. A policy can be deterministic or stochastic. A deterministic policy maps each state to a single action, while a stochastic policy maps each state to a probability distribution over actions.

MDPs are used in many applications, including robotics, game playing, and decision support systems. For example, an autonomous robot might use an MDP to decide which actions to take in order to navigate a room and avoid obstacles. A game-playing agent might use an MDP to decide which moves to make in order to maximize its score. A decision support system might use an MDP to recommend a course of action based on a user's preferences and the current state of the system.

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