Objective Function as Expected Cumulative Reward (Performance Function)

An agent operates in an environment where sequences of events unfold over time. The agent's behavior is described by a policy, denoted as π(a|s), which gives the probability of taking action 'a' when in state 's'. The environment's dynamics are described by a transition function, P(s'|s, a), which gives the probability of moving to the next state 's'' after taking action 'a' in state 's'. The process begins from an initial state, s₀, with a probability of P(s₀).

Consider the following specific two-step sequence of events (a trajectory):

1. The process starts in state s₀.
2. The agent takes action a₀.
3. The environment transitions to state s₁.
4. The agent takes action a₁.
5. The environment transitions to state s₂.

Which expression correctly represents the probability of this entire specific trajectory occurring?

Diagnosing a Faulty Sequence Generation Process

When modeling the generation of a sequence of states and actions as a Markov Decision Process, the probability of transitioning to a new state at any given step depends on the complete history of all states and actions that have occurred since the beginning of the sequence.

Notational Variations in State-Action Sequences (Trajectories)

An agent is generating a sequence by interacting with an environment. For a single time step, starting from state s_t, arrange the following events in the correct logical order.