$$
\begin{array}{l} \log p \left(\boldsymbol {x} ^ {s}, \boldsymbol {y} _ {0} ^ {s}\right) = \log \int p \left(\boldsymbol {x} ^ {s}, \boldsymbol {y} _ {0: T} ^ {s}, \boldsymbol {z} ^ {s}\right) d \boldsymbol {y} _ {1: T} ^ {s} d \boldsymbol {z} ^ {s} \\ \geq \mathbb {E} _ {q \left(\boldsymbol {y} _ {1: T} ^ {s}, \boldsymbol {z} ^ {s} | \boldsymbol {x} ^ {s}, \boldsymbol {y} _ {0} ^ {s}\right)} \left[ \log \frac {p \left(\boldsymbol {y} _ {0 : T} ^ {s} , \boldsymbol {x} ^ {s} , \boldsymbol {z} ^ {s}\right)}{q \left(\boldsymbol {y} _ {1 : T} ^ {s} , \boldsymbol {z} ^ {s} \mid \boldsymbol {x} ^ {s} , \boldsymbol {y} _ {0} ^ {s}\right)} \right] \\ = \mathbb {E} _ {q \left(\boldsymbol {y} _ {1: T} ^ {s}, \boldsymbol {z} ^ {s} | \boldsymbol {x} ^ {s}, \boldsymbol {y} _ {0} ^ {s}\right)} \left[ \log \frac {p \left(\boldsymbol {z} ^ {s}\right) p _ {\phi} \left(\boldsymbol {x} ^ {s} \mid \boldsymbol {z} ^ {s}\right) p _ {\theta} \left(\boldsymbol {y} _ {0 : T} ^ {s} \mid \boldsymbol {x} ^ {s} , \boldsymbol {z} ^ {s}\right)}{q _ {\rho} \left(\boldsymbol {z} ^ {s} \mid \boldsymbol {x} ^ {s} , \boldsymbol {y} _ {0} ^ {s}\right) q \left(\boldsymbol {y} _ {1 : T} ^ {s} \mid \boldsymbol {y} _ {0} , \boldsymbol {z} ^ {s} , \boldsymbol {x} ^ {s}\right)} \right] \\ = \mathbb {E} _ {q \left(\boldsymbol {y} _ {1: T} ^ {s}, \boldsymbol {z} ^ {s} \mid \boldsymbol {x} ^ {s}, \boldsymbol {y} _ {0} ^ {s}\right)} \left[ \log \frac {p (\boldsymbol {z} ^ {s})}{q _ {\rho} \left(\boldsymbol {z} ^ {s} \mid \boldsymbol {y} _ {0} , \boldsymbol {x}\right)} + \log p _ {\phi} (\boldsymbol {x} ^ {s} \mid \boldsymbol {z} ^ {s}) + \log \frac {p _ {\theta} \left(\boldsymbol {y} _ {0 : T} ^ {s} \mid \boldsymbol {x} ^ {s} , \boldsymbol {z} ^ {s}\right)}{q \left(\boldsymbol {y} _ {1: T} ^ {s} \mid \boldsymbol {y} _ {0} ^ {s} , \boldsymbol {z} ^ {s} , \boldsymbol {x} ^ {s}\right)} \right] \\ = \mathbb {E} _ {q _ {\rho} \left(\boldsymbol {z} ^ {s} \mid \boldsymbol {x} ^ {s}, \boldsymbol {y} _ {0} ^ {s}\right)} \left[ \log \frac {p (\boldsymbol {z} ^ {s})}{q _ {\rho} \left(\boldsymbol {z} ^ {s} \mid \boldsymbol {y} _ {0} ^ {s} , \boldsymbol {x} ^ {s}\right)} + \log p _ {\phi} \left(\boldsymbol {x} ^ {s} \mid \boldsymbol {z} ^ {s}\right) \right] + \\ \mathbb {E} _ {q \left(\boldsymbol {y} _ {1: T} ^ {s}, \boldsymbol {z} ^ {s} \mid \boldsymbol {x} ^ {s}, \boldsymbol {y} _ {0} ^ {s}\right)} \left[ \log \frac {p _ {\theta} \left(\boldsymbol {y} _ {0 : T} ^ {s} \mid \boldsymbol {x} ^ {s} , \boldsymbol {z} ^ {s}\right)}{q \left(\boldsymbol {y} _ {1: T} ^ {s} \mid \boldsymbol {y} _ {0} ^ {s} , \boldsymbol {z} ^ {s} , \boldsymbol {x} ^ {s}\right)} \right] \\ = \mathbb {E} _ {q _ {\rho} \left(\boldsymbol {z} ^ {s} \mid \boldsymbol {x} ^ {s}, \boldsymbol {y} _ {0} ^ {s}\right)} \left[ \log p _ {\phi} \left(\boldsymbol {x} ^ {s} \mid \boldsymbol {z} ^ {s}\right) \right] - D _ {K L} \left(q _ {\rho} \left(\boldsymbol {z} ^ {s} \mid \boldsymbol {x} ^ {s}, \boldsymbol {y} _ {0} ^ {s}\right) \| p (\boldsymbol {z} ^ {s})\right) + \\ \underbrace {\mathbb {E} _ {\boldsymbol {z} ^ {s} \sim q _ {\rho} \left(\boldsymbol {z} ^ {s} \mid \boldsymbol {x} ^ {s} , \boldsymbol {y} _ {0} ^ {s}\right)}} _ {\text {①}} \underbrace {\mathbb {E} _ {q \left(\boldsymbol {y} _ {1 : T} ^ {s} \mid \boldsymbol {x} ^ {s} , \boldsymbol {z} ^ {s} , \boldsymbol {y} _ {0} ^ {s}\right)} \left[ \frac {p _ {\theta} \left(\boldsymbol {y} _ {0 : T} ^ {s} \mid \boldsymbol {x} ^ {s} , \boldsymbol {z} ^ {s}\right)}{q \left(\boldsymbol {y} _ {1 : T} ^ {s} \mid \boldsymbol {y} _ {0} ^ {s} , \boldsymbol {z} ^ {s} , \boldsymbol {x} ^ {s}\right)} \right]} _ {\text {②}}. \tag {20} \\ \end{array}
$$