KL-divergence is a non-symmetric measure of the difference between two probability distributions $P$ and $Q$

\[\text{KL}(Q||P)=-\sum\limits_z Q(Z)\log\frac{P(Z|X)}{Q(Z)}\]

The goal is to minimize this function so we have:

\[\text{KL}(Q||P)=-\sum Q(Z)\log \frac{\frac{P(X,Z)}{P(X)}}{\frac{Q(Z)}{1}}\] \[=-\sum Q(Z)\log\frac{P(X,Z)}{Q(Z)}.\frac{1}{P(X)}\] \[=-\sum Q(Z)\left[\log\frac{P(X,Z)}{Q(Z)}-\log P(X)\right]\] \[=-\sum Q(Z)\log\frac{P(X,Z)}{Q(Z)}+\sum Q(Z)\log P(X)\]

since the summation is over $Z$ we can take out $P(X)$ from the second sum and we know that $\sum_z Q(Z)=1$, thus we can write:

\[\log P(X)=\text{KL}(Q||P)+\sum Q(Z)\log\frac{P(X,Z)}{Q(Z)}\]

Note that minimizing the KL-divergence is equivalent to maximizing the second term. The second term in the above equation is called Variational Lower Bound

\[\mathcal{L}=\sum Q(Z)\log\frac{P(X,Z)}{Q(Z)}\]

since KL-divergence is non-negative we can write $\mathcal{L}\leq\log P(X)$ and hence the lower bound name.

\[\mathcal{L}=\sum Q(Z)\log\frac{P(X,Z)}{P(Z)}=\sum Q(Z)\log\frac{P(X|Z)P(Z)}{Q(Z)}\] \[=\sum Q(Z)\left[ \log P(X|Z) + \log\frac{P(Z)}{Q(Z)} \right]\] \[=\sum Q(Z)\log P(X|Z)+\sum Q(Z)\log\frac{P(Z)}{Q(Z)}\] \[=\mathbb{E}_{Q}\left[\log P(X|Z)\right]-\text{KL}(Q(Z|X)||P(Z))\]

Now we can model this with an encoder-decoder (autoencoder) architecture as follows where the encoder is $Q(Z|X)$ and the decoder is modeled by $P(X|Z)$.

We can think of the first part of the objective function as reconstruction error and the second part as a constraint on $P(Z)$ (i.e., $p(Z)$ should be similar to $Q(Z)$). This idea was used in Kingma & Welling, ICLR, 2014 Auto-Encoding Variational Bayes