Improper Priors Can Never Yield Proper Posteriors

Improper priors are widely used in Bayesian statistics, often defended on grounds of convenience or epistemic humility. A common claim is that once evidence is observed, Bayesian updating can yield a proper posterior, from which inferences may proceed as usual (Taraldsen & Lindqvist 2010). I argue this is mistaken. Since a rational Bayesian must assign non-zero probability to the possibility that data are independent of the parameter of interest, multiplication by an improper prior cannot yield a normalizable posterior. Once improper, always improper.