3.4 The warmest-up: Improper Prior

Let \(\pi(\theta)\) denote a prior distribution. If \(\int \pi(\theta) \; d\theta = \infty\), \(\pi(\theta)\) is not a valid probability density, since the prior distribution does not have a finite integral.

 

An example of improper prior is an uninformative location prior: (Jordan 2010)

Consider a probability distribution of density \(f(X - \theta)\) where \(\theta\) is a location parameter that we endow with a prior. A candidate for the prior would be \(\pi(\theta) \propto 1\). If \(\theta\) can take any value in \(\mathbb{R}\), the flat prior is not a probability density since it does not integrate to \(1\).

\[\int_{\mathbb{R}} 1 \; d\theta \rightarrow \infty\]