$$
\begin{array}{l} \mathbb {E} _ {p (\boldsymbol {x})} \mathbb {E} _ {q (z _ {<   l} | \boldsymbol {x})} [ \mathrm {K L} (q (\boldsymbol {z} | \boldsymbol {x}, \boldsymbol {z} _ {<   l}) | | p (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l})) ] \\ = \int q (\boldsymbol {x}) q (\boldsymbol {z} _ {<   l} | \boldsymbol {x}) q (\boldsymbol {z} _ {l} | \boldsymbol {x}, \boldsymbol {z} _ {<   l}) \log \frac {q (\boldsymbol {z} _ {l} | \boldsymbol {x} , \boldsymbol {z} _ {l})}{p (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l})} d \boldsymbol {x} d \boldsymbol {z} _ {l} d \boldsymbol {z} _ {<   l} \\ = \int q (\boldsymbol {x}, \boldsymbol {z} _ {l}, \boldsymbol {z} _ {<   l}) \log \frac {q (\boldsymbol {z} _ {l} | \boldsymbol {x} , \boldsymbol {z} _ {l})}{p (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l})} d \boldsymbol {x} d \boldsymbol {z} _ {l} d \boldsymbol {z} _ {<   l} \\ = \int q (\boldsymbol {x}, \boldsymbol {z} _ {l}, \boldsymbol {z} _ {<   l}) \log \left(\frac {q (\boldsymbol {z} , \boldsymbol {x} | \boldsymbol {z} _ {<   l})}{q (\boldsymbol {x} | \boldsymbol {z} _ {<   l}) q (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l})} \frac {q (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l})}{p (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l})}\right) d \boldsymbol {x} d \boldsymbol {z} _ {l} d \boldsymbol {z} _ {<   l} \\ = \int q \left(\boldsymbol {z} _ {<   l}\right) q \left(\boldsymbol {x}, \boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}\right) \log \frac {q \left(\boldsymbol {z} , \boldsymbol {x} \mid \boldsymbol {z} _ {<   l}\right)}{q \left(\boldsymbol {x} \mid \boldsymbol {z} _ {<   l}\right) q \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}\right)} d \boldsymbol {x} d \boldsymbol {z} _ {l} d \boldsymbol {z} _ {<   l} + \tag {16} \\ \int q (\boldsymbol {x} | \boldsymbol {z} _ {l}, \boldsymbol {z} _ {<   l}) q (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l}) q (\boldsymbol {z} _ {<   l}) \log \frac {q (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l})}{p (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l})} d \boldsymbol {x} d \boldsymbol {z} _ {l} d \boldsymbol {z} _ {<   l} \\ = \int q (\boldsymbol {x}, \boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l}) \log \frac {q (\boldsymbol {z} , \boldsymbol {x} | \boldsymbol {z} _ {<   l})}{q (\boldsymbol {x} | \boldsymbol {z} _ {<   l}) q (\boldsymbol {z} _ {l} | \boldsymbol {z} _ {<   l})} d \boldsymbol {x} d \boldsymbol {z} _ {l} + \\ \int q \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}\right) q \left(\boldsymbol {z} _ {<   l}\right) \log \frac {q \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}\right)}{p \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}\right)} d \boldsymbol {z} _ {l} d \boldsymbol {z} _ {<   l} \\ = H \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}\right) - H \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}, \boldsymbol {x}\right) + \mathbb {E} _ {q \left(\boldsymbol {z} _ {<   l}\right)} \mathrm {K L} \left(q \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l} \right\| p \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}\right)\right) \\ \geq H \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}\right) - H \left(\boldsymbol {z} _ {l} \mid \boldsymbol {z} _ {<   l}, \boldsymbol {x}\right) \\ \end{array}
$$