Chapter 16: Variational Inference and Latent Variables. In the chapter mind map this icon labels **Inference: $p(\mathbf{z}|\mathbf{x})$ Intractability. The discussion below is excerpted and lightly edited from § Posterior Inference & Intractability** in Mathematics for AI and Machine Learning.

In the context of latent variable models, inference refers to the process of reasoning about unobserved (latent) variables $\mathbf z$ given observed data $\mathbf x$. This is the inverse of the generative process: while the model describes how to generate data from latent variables ($\mathbf x$ from $\mathbf z$), inference asks what latent variables are most likely given the observed data ($\mathbf z$ from $\mathbf x$). In Variational Autoencoders (VAEs), the encoder network performs inference by mapping observed data $\mathbf x$ to the parameters of the approximate posterior distribution $q_{\boldsymbol\phi}(\mathbf z | \mathbf x)$.

the book figure visualizes both the generative and inference processes in latent variable models. The left panel shows the latent variables $z$ with a light blue background, representing the prior distribution $p(z)$. The middle panel illustrates both the generative process ($z \rightarrow x$) with a black arrow and the inference process ($x \rightarrow z$) with a red arrow. The right panel shows the observed data $x$, representing the marginal likelihood $p(x)$. This visualization highlights the bidirectional nature of latent variable models: how data is generated from latent variables, and how latent variables are inferred from observed data.

The inference process involves reasoning backward from observed data $x$ to latent variables $z$, which is captured by the posterior distribution $p(z|x)$. This is complemented by the generative process that moves from latent variables to observed data through the likelihood $p(x|z)$.