In a reinforcement learning process, a new policy defined by parameters θ is evaluated using an objective function that relies on data from a reference policy with parameters θ_ref. The objective function is:

J(θ) = E_{τ ~ π_{θ_ref}} [ (Pr_θ(τ) / Pr_{θ_ref}(τ)) * R(τ) ]

Where τ is a trajectory, Pr(τ) is the probability of that trajectory, R(τ) is its total reward, and E_{τ ~ π_{θ_ref}} denotes the expected value over trajectories from the reference policy.

What does this objective function J(θ) simplify to at the specific point where the new policy is identical to the reference policy (i.e., θ = θ_ref)?

Reasoning for Objective Simplification

In a reinforcement learning scenario, the performance of a new policy, defined by parameters θ, is often estimated using an objective function that relies on data collected from a reference policy, defined by parameters θ_ref. This objective function is given by: $J(\theta) = \mathbb{E}_{\tau \sim \pi_{\theta_{\text{ref}}}} \left[ \frac{\text{Pr}_{\theta}(\tau)}{\text{Pr}_{\theta_{\text{ref}}}(\tau)} R(\tau) \right]$ where τ represents a trajectory, Pr(τ) is the probability of that trajectory, and R(τ) is its total reward. Which of the following statements most accurately evaluates the relationship between this objective function, J(θ), and the true expected reward of the reference policy, $\mathbb{E}_{\tau \sim \pi_{\theta_{\text{ref}}}} [R(\tau)]$?