A latent-variable model defines joint density $p_\theta(x,z)=p_\theta(x\mid z)p(z)$ for observation $x$ and latent variable $z$. The posterior is $p_\theta(z\mid x)=p_\theta(x,z)/p_\theta(x)$, where evidence $p_\theta(x)=\int p_\theta(x,z)\,dz$ may be expensive or impossible to compute exactly.
Introduce a tractable approximation $q_\phi(z\mid x)$. The evidence lower bound is $$\mathcal{L}(\theta,\phi;x)=\mathbb{E}_{q_\phi(z\mid x)}[\log p_\theta(x,z)-\log q_\phi(z\mid x)].$$ It can be estimated using samples from $q_\phi$ without directly evaluating the intractable posterior normalization.
The exact identity is $$\log p_\theta(x)=\mathcal{L}(\theta,\phi;x)+D(q_\phi(z\mid x)\Vert p_\theta(z\mid x)).$$ Relative entropy is nonnegative, so $\mathcal{L}\leq\log p_\theta(x)$. The bound becomes tight only when the approximate posterior equals the true posterior within the relevant support.
Rearranging gives a reconstruction-plus-regularization view: $$\mathcal{L}=\mathbb{E}_{q_\phi}[\log p_\theta(x\mid z)]-D(q_\phi(z\mid x)\Vert p(z)).$$ The first term rewards latents that explain the observation. The second keeps the approximate posterior near the prior, enabling generation but potentially limiting information carried by $z$.
For a normal approximation with location $\mu_\phi(x)$ and scale $\sigma_\phi(x)$, write $z=\mu_\phi(x)+\sigma_\phi(x)\odot\epsilon$ with $\epsilon$ sampled from a standard normal distribution. This reparameterization moves randomness into $\epsilon$, allowing pathwise derivatives through location and scale.
Amortized inference uses one network to produce $q_\phi(z\mid x)$ for many observations, trading per-example optimization for speed. The chosen family may miss correlations or multiple modes, and the inference network may not reach the best member of that family. Evaluate bound tightness where possible, latent usage, reconstruction quality, and sensitivity to the approximation family.