Improper priors and improper posteriors

What is a good prior? Actual prior knowledge should be used, but for complex models this often not easily available. The can in the form of symmetry assumptions, and then choice will typically an improper prior. Also more generally, it quite common to choose priors. Motivated by we consider theoretical framework statistics that includes both priors posteriors. Knowledge represented possibly unbounded measure with interpretation as explained Rényi 1955. main mathematical result here constructive proof existence transformation from posterior knowledge. always exists uniquely defined prior, observed data, statistical model. is, be, extension conventional Bayesian inference axioms Kolmogorov. It since novel construction valid also when replacing Kolmogorov conditional probability space. A concrete case based on Markov Chain Monte Carlo simulations data different species tropical butterflies illustrate may appear naturally useful. theory exemplified elementary examples.