Fiducial Inference in Statistical Modeling

Fiducial inference is a framework for assigning a probability distribution to an unknown parameter on the basis of observed data and a statistical model alone, without invoking a prior distribution. Conceived to reconcile aspects of Bayesian and frequentist reasoning, it centres on transforming data via a pivotal quantity so that the parameter enters the distribution explicitly. Early enthusiasm was tempered by concerns over non-uniqueness and lack of general prescription, yet modern advances have revitalised the approach. Generalised fiducial inference constructs a family of data-driven distributions through an inversion of the data-generating equation, while confidence distributions offer a posterior-like summary compatible with frequentist coverage. These methods have found application in areas as diverse as quantitative genetics, reliability analysis, signal processing and clinical trial reporting. By providing exact or approximate probability statements about parameters, fiducial methods facilitate uncertainty quantification, guide decision making and enable the fusion of information from multiple sources. Recent theoretical work has clarified connections between fiducial, likelihood and confidence approaches, leading to more robust algorithms and wider acceptance in practice. The global significance of fiducial inference lies in its ability to deliver interpretable parameter distributions under minimal assumptions, thereby enhancing transparency and reproducibility across disciplines.

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Generalised fiducial methods have been adapted for hypothesis testing in genetic backcross experiments, yielding four distinct p-value constructions that demonstrate superior control of type I error and competitive power in small samples. A real-data study on mouse blood pressure illustrates how these methods accommodate location–scale models and offer both conservative and aggressive testing options, thereby enriching the toolkit for quantitative trait locus analysis.

A 2023 exploration of parameter duality has illuminated the conceptual underpinnings of fiducial and confidence distributions by distinguishing between the fixed true parameter that generated the data and its random counterpart that captures estimation uncertainty. This dual-parameter viewpoint reconciles frequentist and belief-based interpretations, showing that confidence distributions can serve as a bridge between paradigms and resolve apparent conflicts over the nature of probability in inference.

In the context of clinical trials, the notion of a confidence distribution has been proposed as a prior-free analogue to the Bayesian posterior. This approach summarises uncertainty about treatment effects in a single probability distribution, preserves exact frequentist coverage and enables intuitive probability statements without subjective priors. Illustrative case studies demonstrate how confidence distributions can be reported alongside traditional interval estimates to enhance interpretability and decision support in medical research.

Fiducial Inference in Statistical Modeling publication trend

The graph below shows the total number of articles in fiducial inference in statistical modeling across all publications each year (not limited to Nature Index journals).

Technical terms

Fiducial Inference: An approach that derives a probability distribution for a parameter solely from the observed data and model structure, without a prior.

Generalised Fiducial Inference: A modern extension of fiducial ideas that constructs parameter distributions by inverting the data-generating mechanism via simulation or analytic transformation.

Confidence Distribution: A sample-based distribution function on the parameter space that provides a complete summary of inferential uncertainty with frequentist coverage guarantees.

Pivotal Quantity: A function of data and parameter whose probability distribution does not depend on unknown parameters, fundamental to constructing fiducial and confidence distributions.

References

1. Generalized fiducial methods for testing quantitative trait locus effects in genetic backcross studies. Statistical Theory and Related Fields (2021).
2. An Exploration of Parameter Duality in Statistical Inference. Philosophy of Science (2023).
3. Confidence distributions for treatment effects in clinical trials: Posteriors without priors. Statistics in Medicine (2024).