log ⁡ P θ ( x ) = log ⁡ ∫ P θ ( z , x ) d z = log ⁡ ∫ q ϕ ( z ∣ x ) p θ ( z , x ) q ϕ ( z ∣ x ) d z ≥ E q ϕ ( z ∣ x ) log ⁡ p θ ( z , x ) q ϕ ( z ∣ x ) = L \log P_\theta(x)=\log\int P_\theta(z,x)dz\\ =\log\int q_\phi(z|x)\frac{p_\theta(z,x)}{q_\phi(z|x)}dz\\ \ge\mathbb{E}_q\phi(z|x)\log\frac{p_\theta(z,x)}{q_\phi(z|x)}=L logPθ​(x)=log∫Pθ​(z,x)dz=log∫qϕ​(z∣x)qϕ​(z∣x)pθ​(z,x)​dz≥Eq​ϕ(z∣x)logqϕ​(z∣x)pθ​(z,x)​=L

L = E log ⁡ p θ ( x ∣ z ) p θ ( z ) q ϕ ( z ∣ x ) = E q ϕ ( z ∣ x ) log ⁡ P θ ( x ∣ z ) + E q ϕ ( z ∣ x ) log ⁡ p θ ( z ) q ϕ ( z ∣ x ) = E q ϕ ( z ∣ x ) log ⁡ P θ ( x ∣ z ) − K L ( q ϕ ( z ∣ x ) ∣ ∣ p θ ( z ) ) L=\mathbb{E}\log\frac{p\theta(x|z)p_\theta(z)}{q\phi(z|x)}\\ =\mathbb{E}_q\phi(z|x)\log P_\theta(x|z)+\mathbb{E}_q\phi(z|x)\log\frac{p_\theta(z)}{q_\phi(z|x)}\\ =\mathbb{E}_q\phi(z|x)\log P_\theta(x|z)-KL(q_\phi(z|x)||p_\theta(z)) L=Elogqϕ(z∣x)pθ(x∣z)pθ​(z)​=Eq​ϕ(z∣x)logPθ​(x∣z)+Eq​ϕ(z∣x)logqϕ​(z∣x)pθ​(z)​=Eq​ϕ(z∣x)logPθ​(x∣z)−KL(qϕ​(z∣x)∣∣pθ​(z))