$$
\begin{array}{l} \log p (\mathbf {x}; \theta , \beta) \\ = \log \sum_ {k = 1} ^ {K} p (\alpha_ {k}) \int p (\mathbf {x} | \mathbf {z}, \alpha_ {k}; \theta , \beta) p (\mathbf {z}) d \mathbf {z} \\ = \log \sum_ {k = 1} ^ {K} p (\alpha_ {k}) \int p (\mathbf {x} | \mathbf {z}, \alpha_ {k}; \theta , \beta) \frac {p (\mathbf {z})}{q (\mathbf {z} | \mathbf {x} ; \phi)} q (\mathbf {z} | \mathbf {x}; \phi) d \mathbf {z} \\ \geq \sum_ {k = 1} ^ {K} p (\alpha_ {k}) \int q (\mathbf {z} | \mathbf {x}; \phi) \log \left(p (\mathbf {x} | \mathbf {z}, \alpha_ {k}; \theta , \beta) \frac {p (\mathbf {z})}{q (\mathbf {z} | \mathbf {x} ; \phi)}\right) d \mathbf {z} \\ = \frac {1}{K} \sum_ {k = 1} ^ {K} \mathbb {E} _ {q (\mathbf {z} | \mathbf {x}; \phi)} [ \log p (\mathbf {x} | \mathbf {z}, \alpha_ {k}; \theta , \beta) ] - \mathcal {K L} (q (\mathbf {z} | \mathbf {x}; \phi) \| p (\mathbf {z})) \\ = \frac {1}{K} \sum_ {k = 1} ^ {K} \mathbb {E} _ {q (\mathbf {z} | \mathbf {x}; \phi)} \left[ \sum_ {j \in I _ {C}} \log p (\mathbf {x} _ {j} | \mathbf {z}, \alpha_ {k}; \theta_ {j}, \beta) + \sum_ {j \in I _ {D}} \log p (\mathbf {x} _ {j} | \mathbf {z}; \theta_ {j}, \beta) \right] - \mathcal {K L} (q (\mathbf {z} | \mathbf {x}; \phi) \| p (\mathbf {z})) \\ = \mathbb {E} _ {q (\mathbf {z} | \mathbf {x}; \phi)} \left[ \sum_ {j \in I _ {C}} \frac {1}{K} \sum_ {k = 1} ^ {K} \log \frac {\alpha_ {k} (1 - \alpha_ {k})}{\beta} - \rho_ {\alpha_ {k}} \left(\frac {\mathbf {x} _ {j} - D _ {j} (\alpha_ {k} | \mathbf {z} , \theta_ {j})}{\beta}\right) \right] \\ + \mathbb {E} _ {q (\mathbf {z} | \mathbf {x}; \phi)} \left[ \sum_ {j \in I _ {D}} \sum_ {l = 1} ^ {T _ {j}} I (\mathbf {x} _ {j} = l) \cdot \log \pi_ {l} (\mathbf {z}; \theta_ {j}) \right] - \mathcal {K L} (q (\mathbf {z} | \mathbf {x}; \phi) \| p (\mathbf {z})) \\ { = } { \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } ; \phi ) } \left[ - \frac { 1 } { \beta } \cdot \sum _ { j \in I _ { C } } \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \rho _ { \alpha _ { k } } \Bigl ( \mathbf { x } _ { j } - D _ { j } ( \alpha _ { k } | \mathbf { z } , \theta _ { j } ) \Bigr ) \right] + \frac { p } { K } \sum _ { k = 1 } ^ { K } \log \alpha _ { k } ( 1 - \alpha _ { k } ) - p \cdot \log \beta } \\ + \mathbb {E} _ {q (\mathbf {z} | \mathbf {x}; \phi)} \left[ \sum_ {j \in I _ {D}} \sum_ {l = 1} ^ {T _ {j}} I (\mathbf {x} _ {j} = l) \cdot \log \pi_ {l} (\mathbf {z}; \theta_ {j}) \right] - \mathcal {K L} (q (\mathbf {z} | \mathbf {x}; \phi) \| p (\mathbf {z})) \\ { = } { - \frac { 1 } { \beta } \Bigg ( \mathbb { E } _ { q ( \mathbf { z } | \mathbf { x } ; \phi ) } \left[ \sum _ { j \in I _ { C } } \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \rho _ { \alpha _ { k } } \Big ( \mathbf { x } _ { j } - D _ { j } ( \alpha _ { k } | \mathbf { z } , \theta _ { j } ) \Big ) \right] - \beta \frac { p } { K } \sum _ { k = 1 } ^ { K } \log \alpha _ { k } ( 1 - \alpha _ { k } ) + \beta p \cdot \log \beta } \\ \left. - \beta \cdot \mathbb {E} _ {q (\mathbf {z} | \mathbf {x}; \phi)} \left[ \sum_ {j \in I _ {D}} \sum_ {l = 1} ^ {T _ {j}} I (\mathbf {x} _ {j} = l) \cdot \log \pi_ {l} (\mathbf {z}; \theta_ {j}) \right] + \beta \cdot \mathcal {K L} (q (\mathbf {z} | \mathbf {x}; \phi) \| p (\mathbf {z}))\right), \\ \end{array}
$$