An agent executes an identical sequence of states and actions in two different environments, A and B. The agent's policy (π_θ) and a reference policy (π_θ_ref) are also the same in both scenarios. When calculating the approximated policy divergence penalty using the formula Penalty = Σ [log π_θ(a_t|s_t) - log π_θ_ref(a_t|s_t)], the result is identical for both environments. What is the fundamental reason for this?

Calculating Approximated Policy Divergence

Consider the approximated policy divergence penalty formula: $\text{Penalty} = \sum_{t=1}^{T} \log \pi_{\theta}(a_t|s_t) - \sum_{t=1}^{T} \log \pi_{\theta_{\text{ref}}}(a_t|s_t)$ This penalty's value for a fixed trajectory of states and actions is sensitive to changes in the environment's transition dynamics.