Improper priors and improper posteriors

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

Automated summary

The paper discusses the use of symmetry assumptions when actual prior knowledge is not easily accessible for complex models, often resulting in an improper choice of prior. It introduces a theoretical framework for statistics that includes both improper priors and improper posteriors, with a focus on the transformation from prior to posterior knowledge. Through examples like Markov Chain Monte Carlo simulations and data on tropical butterfly species, it illustrates how improper posteriors can naturally occur and be beneficial, extending the conventional Bayesian inference defined by Kolmogorov's axioms to accommodate new constructions based on Renyi's axioms for conditional probability spaces.