Bayes rule is applied to p θ (z t |y <t , z <t ), and we can extract log p θ (z) from the expectation, transfer the expectation term E q φ (zt|y<t,z<t) back to KL-divergence, and rearrange all the terms.
the following holds: Let L(θ, φ; y) represent the last two terms from the right part of Equation 4: Since the first KL-divergence term of Equation 4 is non-negative, we have log p θ (y <t ) ≥ L(θ, φ; y) meaning that L(θ, φ; y) is a lower bound (the objective to be maximized) on the marginal likelihood.