$$
\begin{array}{l} \log p \left(\boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right) = \log \int p \left(\boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0: T} ^ {t}, \boldsymbol {z} ^ {t}\right) d \boldsymbol {y} _ {1: T} ^ {t} d \boldsymbol {z} ^ {t} \\ \geq \mathbb {E} _ {q \left(\boldsymbol {y} _ {1: T} ^ {t}, \boldsymbol {z} ^ {t} \mid \boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right)} \left[ \log \frac {p \left(\boldsymbol {y} _ {0 : T} ^ {t} , \boldsymbol {x} ^ {t} , \boldsymbol {z} ^ {t}\right)}{q \left(\boldsymbol {y} _ {1: T} ^ {t} , \boldsymbol {z} ^ {t} \mid \boldsymbol {x} ^ {t} , \boldsymbol {y} _ {0} ^ {t}\right)} \right] \\ = \mathbb {E} _ {q \left(\boldsymbol {y} _ {1: T} ^ {t}, \boldsymbol {z} ^ {t} \mid \boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right)} \left[ \log \frac {p (\boldsymbol {z} ^ {t}) p _ {\psi} \left(\boldsymbol {x} ^ {t} \mid \boldsymbol {z} ^ {t}\right) p _ {\theta} \left(\boldsymbol {y} _ {0 : T} ^ {t} \mid \boldsymbol {x} ^ {t} , \boldsymbol {z} ^ {t}\right)}{q _ {\rho} \left(\boldsymbol {z} ^ {t} \mid \boldsymbol {x} ^ {t} , \boldsymbol {y} _ {0} ^ {t}\right) q \left(\boldsymbol {y} _ {1: T} ^ {t} \mid \boldsymbol {y} _ {0} , \boldsymbol {z} ^ {t} , \boldsymbol {x} ^ {t}\right)} \right] \\ = \mathbb {E} _ {q \left(\boldsymbol {y} _ {1: T} ^ {t}, \boldsymbol {z} ^ {t} \mid \boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right)} \left[ \log \frac {p (\boldsymbol {z} ^ {t})}{q _ {\rho} \left(\boldsymbol {z} ^ {t} \mid \boldsymbol {y} _ {0} , \boldsymbol {x}\right)} + \log p _ {\psi} \left(\boldsymbol {x} ^ {t} \mid \boldsymbol {z} ^ {t}\right) + \log \frac {p _ {\theta} \left(\boldsymbol {y} _ {0 : T} ^ {t} \mid \boldsymbol {x} ^ {t} , \boldsymbol {z} ^ {t}\right)}{q \left(\boldsymbol {y} _ {1: T} ^ {t} \mid \boldsymbol {y} _ {0} ^ {t} , \boldsymbol {z} ^ {t} , \boldsymbol {x} ^ {t}\right)} \right] \\ = \mathbb {E} _ {q _ {\rho} \left(\boldsymbol {z} ^ {t} | \boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right)} \left[ \log \frac {p (\boldsymbol {z} ^ {t})}{q _ {\rho} \left(\boldsymbol {z} ^ {t} \mid \boldsymbol {y} _ {0} ^ {t} , \boldsymbol {x} ^ {t}\right)} + \log p _ {\psi} \left(\boldsymbol {x} ^ {t} \mid \boldsymbol {z} ^ {t}\right) \right] + \\ \mathbb {E} _ {q \left(\boldsymbol {y} _ {1: T} ^ {t}, \boldsymbol {z} ^ {t} \mid \boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right)} \left[ \log \frac {p _ {\theta} \left(\boldsymbol {y} _ {0 : T} ^ {t} \mid \boldsymbol {x} ^ {t} , \boldsymbol {z} ^ {t}\right)}{q \left(\boldsymbol {y} _ {1: T} ^ {t} \mid \boldsymbol {y} _ {0} ^ {t} , \boldsymbol {z} ^ {t} , \boldsymbol {x} ^ {t}\right)} \right] \\ = \mathbb {E} _ {q _ {\rho} \left(\boldsymbol {z} ^ {t} | \boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right)} \left[ \log p _ {\psi} (\boldsymbol {x} ^ {t} \mid \boldsymbol {z} ^ {t}) \right] - D _ {K L} \left(q _ {\rho} \left(\boldsymbol {z} ^ {t} \mid \boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right) \| p (\boldsymbol {z} ^ {t})\right) + \\ \mathbb {E} _ {\boldsymbol {z} ^ {t} \sim q _ {r h o} \left(\boldsymbol {z} ^ {t} \mid \boldsymbol {x} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right)} \mathbb {E} _ {q \left(\boldsymbol {y} _ {1: T} ^ {t} \mid \boldsymbol {z} ^ {t}, \boldsymbol {y} _ {0} ^ {t}\right)} \left[ \frac {p _ {\theta} \left(\boldsymbol {y} _ {0 : T} ^ {t} \mid \boldsymbol {z} ^ {t}\right)}{q \left(\boldsymbol {y} _ {1 : T} ^ {t} \mid \boldsymbol {y} _ {0} ^ {t} , \boldsymbol {z} ^ {t}\right)} \right] := \mathcal {L} _ {E L B O} ^ {l}. \tag {22} \\ \end{array}
$$