Mathematics for AI and Machine Learning

Foundations for modern AI and machine learning

ELBO

Chapter 14 Probability & information

Chapter 14: Information Theory — Evidence Lower Bound (also appears in Ch. 16)

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ELBO — high-resolution mind-map icon

From the book

Chapter 14: Information Theory. In the chapter mind map this icon labels Evidence Lower Bound. The discussion below is excerpted and lightly edited from § Definition: Evidence Lower Bound in Mathematics for AI and Machine Learning. Related material also appears in Chapter 16 (Variational Inference:ELBO Maximization).

For a probabilistic model with observed data $\mathbf x$, latent variables $\mathbf z$, and parameters $\theta$, the evidence (marginal likelihood) is

where $q(\mathbf z)$ is a variational distribution approximating the true posterior $p_{\boldsymbol\theta}(\mathbf z | \mathbf x)$.

What this drawing shows

What you see. Represents the evidence lower bound used in variational inference and latent-variable models.

In the mind map. Chapter 14 — Evidence Lower Bound. See From the book above for definitions, figures, and worked examples.

Also appears in Ch. 16 (Variational Inference:ELBO Maximization).

Where to read next

Open Chapter 14 companion →

Read the full definitions, figures, and worked examples in Chapter 14: Information Theory — see the mind-map node Evidence Lower Bound.

This concept is also referenced in Chapter 16 (Variational Inference:ELBO Maximization).