Fiducial and Posterior Sampling

TL;DR

This paper generalizes the connection between fiducial and posterior distributions in group models, introduces a broader measure-theoretic framework, and explores implications for invariant measures in loops, offering alternatives to Bayesian sampling.

Contribution

It extends fiducial theory to σ-finite measure spaces and demonstrates new sampling methods, also analyzing invariant measures in non-group structures.

Findings

Fiducial coincides with posterior in group models with Haar prior.

Generalization to σ-finite measure spaces broadens fiducial theory.

Invariant measures may not exist in certain loops, unlike groups.

Abstract

The fiducial coincides with the posterior in a group model equipped with the right Haar prior. This result is here generalized. For this the underlying probability space of Kolmogorov is replaced by a $\sigma$-finite measure space and fiducial theory is presented within this frame. Examples are presented that demonstrate that this also gives good alternatives to existing Bayesian sampling methods. It is proved that the results provided here for fiducial models imply that the theory of invariant measures for groups cannot be generalized directly to loops: There exist a smooth one-dimensional loop where an invariant measure does not exist. Keywords: Conditional sampling, Improper prior, Haar prior, Sufficient statistic, Quasi-group