[Paper Review] Organic fiducial inference

Russell J. Bowater|arXiv (Cornell University)|Jan 23, 2019

Law, Economics, and Judicial SystemsEconomics, Econometrics and Finance11 references3 citations

TL;DR

This paper introduces organic fiducial inference, a generalized framework for statistical inference that extends subjective fiducial methods to discrete and categorical data, incorporates pre-data knowledge via global and local pre-data functions, and avoids reliance on priors or Bayes' theorem. The key contribution is a coherent fiducial approach that handles restricted parameter spaces and incompatible conditional distributions using fiducial densities adjusted by generalized pre-data functions.

ABSTRACT

A substantial generalisation is put forward of the theory of subjective fiducial inference as it was outlined in earlier papers. In particular, this theory is extended to deal with cases where the data are discrete or categorical rather than continuous, and cases where there was important pre-data knowledge about some or all of the model parameters. The system for directly expressing and then handling this pre-data knowledge, which is via what are referred to as global and local pre-data functions for the parameters concerned, is distinct from that which involves attempting to directly represent this knowledge in the form of a prior distribution function over these parameters, and then using Bayes' theorem. In this regard, the individual attributes of what are identified as three separate types of fiducial argument, namely the strong, moderate and weak fiducial arguments, form an integral part of the theory that is developed. Various practical examples of the application of this theory are presented, including examples involving binomial, Poisson and multinomial data. The fiducial distribution functions for the parameters of the models in these examples are interpreted in terms of a generalised definition of subjective probability that was set out previously.

Motivation & Objective

• To extend fiducial inference beyond continuous data to discrete and categorical data, such as binomial and Poisson distributions.
• To develop a method for incorporating substantial pre-data knowledge about model parameters without relying on prior distributions or Bayesian updating.
• To resolve limitations in frequentist and Bayesian approaches when pre-data knowledge restricts parameter spaces or when prior distributions are infeasible to elicit.
• To provide a coherent fiducial framework that handles incompatible conditional distributions through Gibbs sampling and generalized pre-data functions.
• To formalize a generalization of subjective probability that supports fiducial inference under diverse inferential scenarios.

Proposed method

• Uses global and local pre-data functions to encode pre-data knowledge about parameters, distinct from prior distributions.
• Applies a generalized definition of subjective probability to interpret fiducial distributions as post-data knowledge representations.
• Employs fiducial statistics and primary random variables to derive fiducial densities for parameters under various data types.
• Uses Principle 1 and Principle 2 from Section 3.4 to construct fiducial densities in reference scenarios with minimal pre-data knowledge.
• Applies a normalization strategy that treats the generalized pre-data function as a weight to adjust fiducial densities when pre-data knowledge is non-uniform.
• Uses Gibbs sampling to resolve incompatibility between fiducial densities when joint distributions are not consistent.

Experimental results

Research questions

• RQ1 How can fiducial inference be generalized to handle discrete and categorical data, such as binomial and Poisson outcomes, beyond the continuous case?
• RQ2 What is an appropriate method to incorporate substantial pre-data knowledge about model parameters without using a prior distribution?
• RQ3 How can fiducial densities be constructed when the parameter space is restricted, such as when a parameter must exceed a positive lower bound?
• RQ4 In cases where fiducial densities for different parameters are incompatible, how can a coherent joint fiducial distribution be obtained?
• RQ5 What is the role of generalized pre-data functions in adjusting fiducial distributions to reflect post-data knowledge in a way distinct from Bayesian priors?

Key findings

• The theory of organic fiducial inference successfully generalizes fiducial inference to discrete and categorical data, including binomial and Poisson models, by extending fiducial density construction beyond continuous data.
• Pre-data knowledge is encoded via global and local pre-data functions, which act as weight functions to adjust fiducial densities without requiring a prior distribution.
• When fiducial densities for multiple parameters are incompatible, the Gibbs sampling algorithm can be used to construct a consistent joint fiducial distribution.
• The method avoids reliance on Bayes' theorem and provides a coherent alternative to both frequentist and Bayesian inference in scenarios with restricted parameter spaces.
• The generalized pre-data function is shown to function as a weight that adjusts fiducial densities derived from reference scenarios, enabling flexible and interpretable inference under diverse pre-data knowledge.
• The approach maintains coherence and interpretability by grounding fiducial distributions in a generalized subjective probability framework, distinct from Bayesian priors.