We aim for some probability distribution \[\theta \sim \pi(\theta), \, \theta \in \Theta\]

We want some summaries of this distribution \[ E_{\pi}g(\theta) = \int_{\Theta} g(\theta) \pi(\theta) d\theta \]

In many cases \(\pi(\theta)\) is the posterior distribution from some Bayesian model: \[ \pi(\theta) = p(\theta|y) \propto p(y|\theta) p(\theta)\]