It represents the set of possible unkown states of nature.
It is where the data \(x\) lies.
Often indexed by \(\theta\) \[f(x; \theta)\]
Set of all actions or decisions. We use \(a\) to refer to a particular action.
\(L: \Theta \times \mathcal{A} \rightarrow \mathbb{R}\)
If we choose the action \(a \in \mathcal{A}\) when the parameter is \(\theta \in \Theta\), we incur a loss \(L(\theta,a)\).
\(d: \mathcal{X} \rightarrow \mathcal{A}\)
\(R(\theta,d) = E_{\theta} [L(\theta,d(X))] = \int L(\theta,d(X)) f(x; \theta) dx\)
\(\overline{R}(\widehat{\theta}) = \sup_\theta R(\theta,\widehat{\theta})\)
\(r(f,\widehat{\theta}) = \int R(\theta,\widehat{\theta}) f(\theta) d\theta\) where \(f(\theta)\) is a prior for \(\theta\).