(not quite covered in the chapter)
Instead of using ELBO, we may choose to estimate the multi-dimensional integral directly using MCMC. MCMC can be combined with many other techniques (for instance, MCMC-EM is a thing).
Unlike what Chapter 19 focuses on, there are other approximation techniques for dealing with intractable/hard-to-compute posteriors.
Here we include some sampling-based methods.
Slides: Approximate Inference in Bayes Nets - Sampling based methods
Slides: Approximate inference — touches on sampling methods after talking about Variational Inference
MCMC - Wikipedia
Markov chain Monte Carlo - Wikipedia
Slides: Generalized EM with Gibbs Sampling