Bayesian inference

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

Treat parameters as random and update a prior density into a posterior with Bayes' rule. The normalizing integral is intractable in general, so practice uses MCMC (sampling) or variational inference (optimization).

Formulas

$p(\theta\mid D) = \frac{p(D\mid\theta)\,p(\theta)}{\int p(D\mid\theta')\,p(\theta')\,\dd\theta'}$; $\log p(D) \ge \E_{q}\big[\log p(D,\theta) - \log q(\theta)\big]$
evidence lower bound (ELBO)

The mathematics behind it

Probability density function★★★★★fundamental
Priors, likelihoods and posteriors over continuous parameters are densities.
Improper integrals★★★★★fundamental
The evidence $p(D) = \int p(D\mid\theta)\,p(\theta)\,\dd\theta$ is an integral over the whole parameter space.
Numerical integration (quadrature)★★★★★frequent
Low-dimensional posteriors and marginal likelihoods can be computed by quadrature.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

ℒ AI and machine learning

Generative models★★★★★

What depends on it

Generative models

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.