$$
\begin{array}{l} \log p _ {\theta} \left(Y _ {T} ^ {0: t} \mid X _ {T} ^ {0: t}, \mathcal {C}\right) \quad (\text {L o g - l i k e l i h o o d o f e v i d e n c e}) (10a) \\ = \log p _ {\theta} \left(Y _ {\mathcal {T}} ^ {0: t} \mid X _ {\mathcal {T}} ^ {0: t}, \mathcal {C}\right) \int p _ {\theta} \left(z ^ {G} \mid X _ {\mathcal {T}} ^ {0: t}, Y _ {\mathcal {T}} ^ {0: t}, \mathcal {C}\right) d z ^ {G} \quad \left(\because \int p _ {\theta} \left(z ^ {G} \mid \mathcal {T}, \mathcal {C}\right) d z ^ {G} = 1\right) (10b) \\ = \int p _ {\theta} \left(z ^ {G} \mid X _ {\mathcal {T}} ^ {0: t}, Y _ {\mathcal {T}} ^ {0: t}, \mathcal {C}\right) \left(\log p _ {\theta} \left(Y _ {\mathcal {T}} ^ {0: t} \mid X _ {\mathcal {T}} ^ {0: t}, \mathcal {C}\right)\right) d z ^ {G} \quad (\text {I n t e g r a t e o v e r t h e l o g - l i k e l i h o o d}) (10c) \\ = \mathbb {E} _ {q _ {\phi} (z G | _ {\mathcal {T}})} \left[ \log p _ {\theta} \left(Y _ {\mathcal {T}} ^ {0: t} \mid X _ {\mathcal {T}} ^ {0: t}, \mathcal {C}\right) \right] \quad (\text {B y}) \\ = \mathbb {E} _ {q _ {\phi} (z ^ {G} | \mathcal {T})} \left[ \log \frac {p _ {\theta} \left(Y _ {\mathcal {T}} ^ {0 : t} , z ^ {G} \mid X _ {\mathcal {T}} ^ {0 : t} , \mathcal {C}\right)}{p _ {\theta} \left(z ^ {G} \mid X _ {\mathcal {T}} ^ {0 : t} , Y _ {\mathcal {T}} ^ {0 : t} , \mathcal {C}\right)} \right] \quad (\text {R e - i n t r o d u c e} z ^ {G} \text {b y C h a i n r u l e}) (10e) \\ = \mathbb {E} _ {q _ {\phi} (z ^ {G} | \mathcal {T})} \left[ \log \frac {p _ {\theta} \left(Y _ {\mathcal {T}} ^ {0 : t} \mid X _ {\mathcal {T}} ^ {0 : t} , \mathcal {C} , z ^ {G}\right) p _ {\theta} \left(z ^ {G} \mid X _ {\mathcal {T}} ^ {0 : t} , \mathcal {C}\right)}{p _ {\theta} \left(z ^ {G} \mid \mathcal {T}\right)} \right] \quad (\text {C h a i n r u l e o f p r o b a b i l i t y}; \mathcal {C} \subset \mathcal {T}) (10f) \\ = \mathbb {E} _ {q _ {\phi} (z ^ {G} | \mathcal {T})} \left[ \log \frac {p _ {\theta} \left(Y _ {\mathcal {T}} ^ {0 : t} \mid X _ {\mathcal {T}} ^ {0 : t} , \mathcal {C} , z ^ {G}\right) p _ {\theta} \left(z ^ {G} \mid X _ {\mathcal {T}} ^ {0 : t} , \mathcal {C}\right) q _ {\phi} \left(z ^ {G} \mid \mathcal {T}\right)}{p _ {\theta} \left(z ^ {G} \mid \mathcal {T}\right) q _ {\phi} \left(z ^ {G} \mid \mathcal {T}\right)} \right] \quad (\text {E q u i v a l e n t f r a c t i o n}) (10g) \\ = \mathbb {E} _ {q _ {\phi} (z ^ {G} | \mathcal {T})} \left[ \log p _ {\theta} (Y _ {\mathcal {T}} ^ {0: t} | X _ {\mathcal {T}} ^ {0: t}, \mathcal {C}, z ^ {G}) \right] \\ + \mathbb {E} _ {q _ {\phi} (z G | \mathcal {T})} \left[ \log \frac {p _ {\theta} (z ^ {G} | \mathcal {C})}{q _ {\phi} (z ^ {G} | \mathcal {T})} \right] + \mathbb {E} _ {q _ {\phi} (z G | \mathcal {T})} \left[ \log \frac {q _ {\phi} (z ^ {G} | \mathcal {T})}{p _ {\theta} (z ^ {G} | \mathcal {T})} \right] \quad (\text {S p l i t} \\ = \mathbb {E} _ {q _ {\phi} (z ^ {G} | \mathcal {T})} \left[ \log p _ {\theta} \big (Y _ {\mathcal {T}} ^ {0: t} | X _ {\mathcal {T}} ^ {0: t}, \mathcal {C}, z ^ {G} \big) \right] \\ - D _ {\mathrm {K L}} \left(q _ {\phi} \left(z ^ {G} | \mathcal {T}\right) \| p _ {\theta} \left(z ^ {G} | \mathcal {C}\right)\right) + D _ {\mathrm {K L}} \left(q _ {\phi} \left(z ^ {G} | \mathcal {T}\right) \| p _ {\theta} \left(z ^ {G} | \mathcal {T}\right)\right) \quad (\text {B y}) \\ \geq \mathbb {E} _ {q _ {\phi} (z ^ {G} | \mathcal {T})} \left[ \log p _ {\theta} \left(Y _ {\mathcal {T}} ^ {0: t} \mid X _ {\mathcal {T}} ^ {0: t}, \mathcal {C}, z ^ {G}\right) \right] - D _ {\mathrm {K L}} \left(q _ {\phi} \left(z ^ {G} | \mathcal {T}\right) \| p _ {\theta} \left(z ^ {G} | \mathcal {C}\right)\right), \quad (\because \mathrm {K L} \text {d i v e r g e n c e} \geq 0) (10j) \\ \end{array}
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